POO

5 files changed, 4767 insertions, 0 deletions

diff --git a/fvv1/n.lyx b/fvv1/n.lyx
new file mode 100644
index 0000000..0e1cf08
--- /dev/null
+++ b/fvv1/n.lyx
@@ -0,0 +1,205 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
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+\language spanish
+\language_package default
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+\cite_engine basic
+\cite_engine_type default
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+\justification true
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+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 0.2cm
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+\end_header
+
+\begin_body
+
+\begin_layout Title
+Funciones de varias variables I
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2018}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Antonio Avilés.
+\end_layout
+
+\begin_layout Itemize
+FVV1, Luis Oncina (2016–17).
+\end_layout
+
+\begin_layout Chapter
+Normas y convergencia
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Derivadas y diferenciales
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Dobles diferenciales
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Regiones de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv1/n1.lyx b/fvv1/n1.lyx
new file mode 100644
index 0000000..41aa455
--- /dev/null
+++ b/fvv1/n1.lyx
@@ -0,0 +1,1914 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
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+\font_sf_scale 100 100
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+\use_dash_ligatures true
+\graphics default
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+\output_sync 0
+\bibtex_command default
+\index_command default
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+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
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+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $E$
+\end_inset
+
+ un 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+-espacio vectorial, una 
+\series bold
+norma
+\series default
+ es una aplicación 
+\begin_inset Formula $\Vert\cdot\Vert:E\rightarrow\mathbb{R}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x,y\in E,\lambda\in\mathbb{R}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert x\Vert\geq0\land(\Vert x\Vert=0\iff x=0)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert x+y\Vert\leq\Vert x\Vert+\Vert y\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert\lambda x\Vert=|\lambda|\Vert x\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El par 
+\begin_inset Formula $(E,\Vert\cdot\Vert)$
+\end_inset
+
+ es un 
+\series bold
+espacio normado
+\series default
+.
+ Llamamos 
+\series bold
+distancia asociada a la norma
+\series default
+ a 
+\begin_inset Formula $d(x,y):=\Vert x-y\Vert$
+\end_inset
+
+.
+ Dos normas son equivalentes si sus distancias lo son.
+\end_layout
+
+\begin_layout Standard
+Ejemplos de normas en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ son las dadas por 
+\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{p}:=\sqrt[p]{\sum_{i=1}^{n}|x_{i}|^{p}}$
+\end_inset
+
+ y
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{\infty}:=\max\{|x_{i}|\}_{i=1}^{n}$
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $V:={\cal C}[a,b]:=\{f:[a,b]\rightarrow\mathbb{R}\text{ continua}\}$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert f\Vert_{\infty}:=\sup\{|f(x)|\}_{x\in[a,b]}$
+\end_inset
+
+ es un espacio normado.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Comment
+status open
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $f\in V$
+\end_inset
+
+, existe al menos un 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\Vert f\Vert_{\infty}\geq|f(x)|\geq0$
+\end_inset
+
+, y 
+\begin_inset Formula $\Vert f\Vert_{\infty}=0\iff\forall x\in[a,b],f(x)=0\iff f=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert f+g\Vert_{\infty}=\sup\{|f(x)+g(x)|\}_{x\in[a,b]}\leq\sup\{|f(x)|+|g(x)|\}_{x\in[a,b]}\leq\sup\{|f(x)|\}_{x\in[a,b]}+\sup\{|g(x)|\}_{x\in[a,b]}=\Vert f\Vert_{\infty}+\Vert g\Vert_{\infty}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $\lambda\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert\lambda f\Vert_{\infty}=\sup\{|\lambda f(x)|\}_{x\in[a,b]}=|\lambda|\sup\{|f(x)|\}_{x\in[a,b]}=|\lambda|\Vert f\Vert_{\infty}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{TEM}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:(X,d)\rightarrow(Y,d')$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $p$
+\end_inset
+
+ [...] si y sólo si 
+\begin_inset Formula $\forall\varepsilon>0,\exists\delta>0:\forall x\in X,(d(x,p)<\delta\implies d'(f(x),f(p))<\varepsilon)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Definimos la norma de una aplicación 
+\begin_inset Formula $L:(E,\Vert\cdot\Vert)\rightarrow(F,\Vert\cdot\Vert')$
+\end_inset
+
+ como 
+\begin_inset Formula $\Vert L\Vert:=\Vert L\Vert_{\Vert\cdot\Vert}^{\Vert\cdot\Vert'}:=\sup\{\Vert L(x)\Vert'\}_{x\in E,\Vert x\Vert\leq1}$
+\end_inset
+
+, y tenemos como 
+\series bold
+teorema
+\series default
+ que 
+\begin_inset Formula $L$
+\end_inset
+
+ es continua si y sólo si 
+\begin_inset Formula $\Vert L\Vert<+\infty$
+\end_inset
+
+, y entonces es uniformemente continua.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $L$
+\end_inset
+
+ continua en 0, es decir, 
+\begin_inset Formula $\forall\varepsilon>0,\exists\delta>0:\forall y\in E,(\Vert y\Vert<\delta\implies\Vert L(y)\Vert'<\varepsilon)$
+\end_inset
+
+.
+ Fijado 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+, sea 
+\begin_inset Formula $\Vert z\Vert\leq1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\Vert\frac{\delta}{2}z\Vert<\delta$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert L(\frac{\delta}{2}z)\Vert'<\varepsilon$
+\end_inset
+
+, luego 
+\begin_inset Formula $\Vert L(z)\Vert'<\frac{2\varepsilon}{\delta}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert L\Vert<\frac{2\varepsilon}{\delta}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Veamos primero que 
+\begin_inset Formula $\Vert L\Vert<+\infty\implies\forall x\in E,\Vert L(x)\Vert'\leq\Vert L\Vert\Vert x\Vert$
+\end_inset
+
+.
+ En efecto, para 
+\begin_inset Formula $\Vert x\Vert=1$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert L(x)\Vert'\leq\sup\{\Vert L(y)\Vert\}_{\Vert y\Vert\leq1}=\Vert L\Vert=\Vert L\Vert\Vert x\Vert$
+\end_inset
+
+, y para cualquier otra 
+\begin_inset Formula $x$
+\end_inset
+
+ basta dividir entre la norma.
+ Ahora bien, dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, tomando 
+\begin_inset Formula $\delta:=\frac{\varepsilon}{\Vert L\Vert+1}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\Vert y-x\Vert<\delta\implies\Vert L(y)-L(x)\Vert'=\Vert L(y-x)\Vert'\leq\Vert L\Vert\Vert y-x\Vert<\Vert L\Vert\delta=\frac{\Vert L\Vert\varepsilon}{\Vert L\Vert+1}<\varepsilon$
+\end_inset
+
+.
+ Pero como 
+\begin_inset Formula $\delta$
+\end_inset
+
+ no depende de 
+\begin_inset Formula $x$
+\end_inset
+
+, 
+\begin_inset Formula $L$
+\end_inset
+
+ es uniformemente continua.
+\end_layout
+
+\begin_layout Section
+Equivalencia de normas
+\end_layout
+
+\begin_layout Standard
+Dos normas 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert\cdot\Vert'$
+\end_inset
+
+ son equivalentes si y sólo si 
+\begin_inset Formula $\exists\alpha,\beta>0:\forall x\in E,\alpha\Vert x\Vert\leq\Vert x\Vert'\leq\beta\Vert x\Vert$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $L:=id_{E}:(E,\Vert\cdot\Vert)\rightarrow(E,\Vert\cdot\Vert')$
+\end_inset
+
+ y 
+\begin_inset Formula $L':=L^{-1}$
+\end_inset
+
+, entonces 
+\begin_inset Formula ${\cal T}_{\Vert\cdot\Vert}={\cal T}_{\Vert\cdot\Vert'}$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $L$
+\end_inset
+
+ y 
+\begin_inset Formula $L'$
+\end_inset
+
+ son continuas, pues 
+\begin_inset Formula $L\text{ es continua}\iff\forall A\in{\cal T}_{\Vert\cdot\Vert'},A\in{\cal T}_{\Vert\cdot\Vert}\iff{\cal T}_{\Vert\cdot\Vert'}\subseteq{\cal T}_{\Vert\cdot\Vert}$
+\end_inset
+
+, y el otro contenido es análogo.
+ Entonces:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $L$
+\end_inset
+
+ es continua 
+\begin_inset Formula $\Vert L\Vert<+\infty$
+\end_inset
+
+, luego 
+\begin_inset Formula $\Vert x\Vert'=\Vert L(x)\Vert'\leq\Vert L\Vert\Vert x\Vert\overset{\beta:=\Vert L\Vert}{=}\beta\Vert x\Vert$
+\end_inset
+
+.
+ La otra cota se hace de forma análoga.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si existe 
+\begin_inset Formula $\beta$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in E,\Vert x\Vert'\leq\beta\Vert x\Vert$
+\end_inset
+
+, en particular se cumple para 
+\begin_inset Formula $\Vert x\Vert\leq1$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\Vert L(x)\Vert=\Vert x\Vert'\leq\beta\Vert x\Vert$
+\end_inset
+
+, luego 
+\begin_inset Formula $\Vert L\Vert=\sup\{\Vert x\Vert'\}_{\Vert x\Vert\leq1}\leq\beta<+\infty$
+\end_inset
+
+ y 
+\begin_inset Formula $L$
+\end_inset
+
+ es continua.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{TEM}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Las métricas 
+\begin_inset Formula $d_{E}$
+\end_inset
+
+, 
+\begin_inset Formula $d_{T}$
+\end_inset
+
+ y 
+\begin_inset Formula $d_{\infty}$
+\end_inset
+
+ [...] son equivalentes, [...].
+ 
+\series bold
+Demostración:
+\series default
+ Se deduce de que 
+\begin_inset Formula $\frac{1}{n}d_{T}(x,y)\leq d_{\infty}(x,y)\leq d_{T}(x,y)$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{1}{\sqrt{n}}d_{E}(x,y)\leq d_{\infty}(x,y)\leq d_{E}(x,y)$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Todo cerrado 
+\begin_inset Formula $C$
+\end_inset
+
+ de un compacto 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ es compacto.
+ [...] En [...] 
+\begin_inset Formula $(\mathbb{R},{\cal T}_{u})$
+\end_inset
+
+ [...] todo subespacio cerrado y acotado es compacto.
+ [...] Todo subespacio compacto 
+\begin_inset Formula $K$
+\end_inset
+
+ de un espacio métrico 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ es [cerrado y] acotado.
+ [...]Si 
+\begin_inset Formula $f:(X,{\cal T})\rightarrow(Y,{\cal T}')$
+\end_inset
+
+ es continua y 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ es compacto entonces 
+\begin_inset Formula $f(X)$
+\end_inset
+
+ es compacto.
+ [...] 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ es 
+\series bold
+compacto 
+\series default
+[...] si toda sucesión admite una subsucesión convergente.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Toda norma 
+\begin_inset Formula $\Vert\cdot\Vert:(E,\Vert\cdot\Vert)\rightarrow\mathbb{R}$
+\end_inset
+
+ es uniformemente continua.
+ 
+\series bold
+Demostración:
+\series default
+ Fijado 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ y tomando 
+\begin_inset Formula $\delta:=\varepsilon$
+\end_inset
+
+, si 
+\begin_inset Formula $\Vert x-y\Vert<\delta$
+\end_inset
+
+ entonces usando que 
+\begin_inset Formula $|\Vert x\Vert-\Vert y\Vert|\leq\Vert x-y\Vert$
+\end_inset
+
+, lo que se deduce de 
+\begin_inset Formula $\Vert x\Vert\leq\Vert x-y\Vert+\Vert y\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert y\Vert\leq\Vert y-x\Vert+\Vert x\Vert$
+\end_inset
+
+, obtenemos que 
+\begin_inset Formula $|\Vert x\Vert-\Vert y\Vert|<\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, todas las normas son equivalentes.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\Vert x\Vert\leq C\Vert x\Vert_{1}]$
+\end_inset
+
+ 
+\begin_inset Formula $\Vert x\Vert=\Vert\sum x_{i}\vec{e}_{i}\Vert\leq\sum|x_{i}|\Vert\vec{e}_{i}\Vert\leq\max\{\Vert\vec{e}_{i}\Vert\}\sum|x_{i}|=\max\{\Vert\vec{e}_{i}\Vert\}\Vert x\Vert_{1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\Vert x\Vert_{1}\leq D\Vert x\Vert]$
+\end_inset
+
+ Tomamos 
+\begin_inset Formula $(\mathbb{R}^{n},\Vert\cdot\Vert_{1})\overset{id}{\rightarrow}(\mathbb{R}^{n},\Vert\cdot\Vert)\overset{\Vert\cdot\Vert}{\rightarrow}\mathbb{R}$
+\end_inset
+
+, que es continua por ser composición de dos funciones continuas (la identidad
+ es continua por la otra cota y la demostración del teorema anterior), entonces
+ 
+\begin_inset Formula $S:=\{x\in\mathbb{R}^{n}:\Vert x\Vert_{1}=1\}$
+\end_inset
+
+ es cerrado dentro del compacto 
+\begin_inset Formula $\overline{B}(0,1)$
+\end_inset
+
+, luego es compacto y como la función dada es continua, 
+\begin_inset Formula $\Vert\cdot\Vert(S)$
+\end_inset
+
+ es compacto y alcanza su máximo y su mínimo.
+ Sea ahora 
+\begin_inset Formula $\mu:=\min\{\Vert x\Vert\}_{x\in S}>0$
+\end_inset
+
+ (pues 
+\begin_inset Formula $0\notin S$
+\end_inset
+
+), si 
+\begin_inset Formula $\Vert x\Vert_{1}=1$
+\end_inset
+
+ para un cierto 
+\begin_inset Formula $x\in\mathbb{R}^{n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\Vert x\Vert\geq\mu$
+\end_inset
+
+, luego 
+\begin_inset Formula $x\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $\left\Vert \frac{x}{\Vert x\Vert_{1}}\right\Vert =1$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\left\Vert \frac{x}{\Vert x\Vert_{1}}\right\Vert \geq\mu$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert x\Vert\geq\mu\Vert x\Vert_{1}$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\Vert x\Vert_{1}\leq\frac{1}{\mu}\Vert x\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Tenemos pues que toda 
+\begin_inset Formula $T:(\mathbb{R}^{m},\Vert\cdot\Vert)\rightarrow(\mathbb{R}^{n},\Vert\cdot\Vert')$
+\end_inset
+
+ lineal es continua
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues equivale a una multiplicación por una matriz en 
+\begin_inset Formula $M_{n\times m}(\mathbb{R})$
+\end_inset
+
+, que es continua con la norma euclídea y por tanto en todas las demás
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Convergencia
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(X,{\cal T})$
+\end_inset
+
+ un espacio topológico e 
+\begin_inset Formula $(Y,d)$
+\end_inset
+
+ un espacio métrico, una sucesión de funciones 
+\begin_inset Formula $(f_{n}:(X,{\cal T})\rightarrow(Y,d))_{n}$
+\end_inset
+
+ 
+\series bold
+converge puntualmente
+\series default
+ a 
+\begin_inset Formula $f:(X,{\cal T})\rightarrow(Y,d)$
+\end_inset
+
+ si para todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, 
+\begin_inset Formula $f_{n}(x)\rightarrow f(x)$
+\end_inset
+
+, y converge 
+\series bold
+uniformemente
+\series default
+ a 
+\begin_inset Formula $f$
+\end_inset
+
+ si 
+\begin_inset Formula $d_{\infty}(f_{n},f)\rightarrow0$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $x_{0}\in(X,{\cal T})$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{n}:X\rightarrow\mathbb{R})_{n}$
+\end_inset
+
+ una sucesión de funciones continuas en 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ que converge uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Fijado un 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existe un 
+\begin_inset Formula $n_{0}\in\mathbb{N}$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $n\geq n_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $|f_{n}(x)-f(x)|<\frac{\varepsilon}{3}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $f_{n}$
+\end_inset
+
+ es continua, existe 
+\begin_inset Formula ${\cal V}\in{\cal E}(x_{0})$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $x\in{\cal V}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $|f_{n}(x)-f_{n}(x_{0})|<\frac{\varepsilon}{3}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $|f(x)-f(x_{0})|\leq|f(x)-f_{n}(x)|+|f_{n}(x)-f_{n}(x_{0})|+|f_{n}(x_{0})-f(x_{0})|<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}=\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+El límite puntual de funciones continuas no es necesariamente continua,
+ ni siquera en funciones de 
+\begin_inset Formula $(a,b)$
+\end_inset
+
+ a 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+, y como contraejemplo tenemos 
+\begin_inset Formula $(f_{n}:(-1,1)\rightarrow\mathbb{R})_{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f_{n}(x)=\begin{cases}
+0 & \text{si }x\leq0\\
+nx & \text{si }0<x<\frac{1}{n}\\
+1 & \text{si }x\geq\frac{1}{n}
+\end{cases}$
+\end_inset
+
+, que converge a 
+\begin_inset Formula $f(x)=\begin{cases}
+0 & \text{si }x\leq0\\
+1 & \text{si }x>0
+\end{cases}$
+\end_inset
+
+.
+ Asimismo, el límite uniforme de funciones derivables no es necesariamente
+ derivable, y como contraejemplo tenemos 
+\begin_inset Formula $f_{n}(x)=\sqrt{x^{2}+\frac{1}{n}}$
+\end_inset
+
+, que converge a 
+\begin_inset Formula $f(x)=|x|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+La función 
+\begin_inset Formula $\sum_{n}\frac{\cos(7^{n}\pi x)}{2^{n}}$
+\end_inset
+
+ es continua en todos los puntos (
+\begin_inset Formula $\Vert\frac{\cos(7^{n}\pi x)}{2^{n}}\Vert_{\infty}=\frac{1}{2^{n}}\implies\Vert\sum_{n}\frac{\cos(7^{n}\pi x)}{2^{n}}\Vert<+\infty$
+\end_inset
+
+, con lo que la función es continua por ser la suma de una sucesión convergente
+ de funciones continuas) pero no es derivable en ninguno.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{FUVR1}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lim_{n}x_{n}=\pm\infty$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\lim_{n}\left(1+\frac{1}{x_{n}}\right)^{x_{n}}=e$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{n}\left(1-\frac{1}{x_{n}}\right)^{x_{n}}=e^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si existe 
+\begin_inset Formula $\lim_{n}\frac{z_{n+1}}{z_{n}}=w\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $|w|<1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\lim_{n}z_{n}=0$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $b>0$
+\end_inset
+
+, 
+\begin_inset Formula $c>1$
+\end_inset
+
+ y 
+\begin_inset Formula $d>0$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\log n\ll n^{b}\ll c^{n}\ll n^{dn}
+\]
+
+\end_inset
+
+Si además 
+\begin_inset Formula $d\geq1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $c^{n}\ll n!\ll n^{dn}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\lim_{n}x_{n}=0$
+\end_inset
+
+ con 
+\begin_inset Formula $0<|x_{n}|<1$
+\end_inset
+
+, entonces:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\log(1+x_{n})\sim x_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $e^{x_{n}}-1\sim x_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\lim_{n}x_{n}=1$
+\end_inset
+
+ con 
+\begin_inset Formula $x_{n}\neq1$
+\end_inset
+
+ y 
+\begin_inset Formula $\lim_{n}y_{n}=\pm\infty$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\lim_{n}x_{n}^{y_{n}}=e^{\lim_{n}y_{n}(x_{n}-1)}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\lim_{n}x_{n}=0$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{n}\neq0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sin x_{n}\sim x_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterios de Stolz:
+\series default
+ Si 
+\begin_inset Formula $(a_{n})_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(b_{n})_{n}$
+\end_inset
+
+ son sucesiones de reales tales que 
+\begin_inset Formula $(b_{n})_{n}$
+\end_inset
+
+ es estrictamente creciente o decreciente y bien 
+\begin_inset Formula $\lim_{n}a_{n}=\lim_{n}b_{n}=0$
+\end_inset
+
+, bien 
+\begin_inset Formula $\lim_{n}b_{n}=\infty$
+\end_inset
+
+, si existe 
+\begin_inset Formula $\lim_{n}\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}=L\in\overline{\mathbb{R}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\lim_{n}\frac{a_{n}}{b_{n}}=L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como consecuencia:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(a_{n})_{n}$
+\end_inset
+
+ converge, entonces
+\begin_inset Formula 
+\[
+\lim_{n}\frac{a_{1}+\dots+a_{n}}{n}=\lim_{n}a_{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(a_{n})_{n}$
+\end_inset
+
+ converge y 
+\begin_inset Formula $a_{n}>0$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\lim_{n}\sqrt[n]{a_{1}\cdots a_{n}}=\lim_{n}a_{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $a_{n}>0$
+\end_inset
+
+ y existe 
+\begin_inset Formula $\lim_{n}\frac{a_{n}}{a_{n-1}}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\lim_{n}\sqrt[n]{a_{n}}=\lim_{n}\frac{a_{n}}{a_{n-1}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{FUVR1}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+condición de Cauchy
+\series default
+ nos dice que 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ es convergente si y sólo si 
+\begin_inset Formula $\forall\varepsilon>0,\exists n_{0}\in\mathbb{N}:\forall p,q\in\mathbb{N},(n_{0}\leq p\leq q\implies|a_{p+1}+\dots+a_{q}|<\varepsilon)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Si 
+\begin_inset Formula $S_{n}$
+\end_inset
+
+ converge, entonces 
+\begin_inset Formula $\lim_{n}a_{n}=0$
+\end_inset
+
+.
+ [...] La convergencia de una serie no se altera modificando un número finito
+ de términos de esta.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dada una serie 
+\begin_inset Formula $\sum_{n=1}^{\infty}a_{n}$
+\end_inset
+
+ de términos 
+\begin_inset Formula $a_{n}\geq0$
+\end_inset
+
+, esta es convergente si y sólo si la sucesión de sumas parciales es acotada
+ [...].
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterios de comparación:
+\end_layout
+
+\begin_layout Enumerate
+Dadas 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sum_{n}b_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n},b_{n}\geq0$
+\end_inset
+
+, si existe 
+\begin_inset Formula $M>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $a_{n}\leq Mb_{n}\forall n$
+\end_inset
+
+, entonces la convergencia de 
+\begin_inset Formula $\sum_{n=1}^{\infty}b_{n}$
+\end_inset
+
+ implica la de 
+\begin_inset Formula $\sum_{n=1}^{\infty}a_{n}$
+\end_inset
+
+ [...].
+\end_layout
+
+\begin_layout Enumerate
+Dadas 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sum_{n}b_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n},b_{n}>0$
+\end_inset
+
+ y existe 
+\begin_inset Formula $l:=\lim_{n}\frac{a_{n}}{b_{n}}\in\mathbb{R}\cup\{+\infty\}$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $0<l<\infty$
+\end_inset
+
+, ambas series tienen el mismo carácter.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $l=0$
+\end_inset
+
+ entonces la convergencia de 
+\begin_inset Formula $\sum_{n}b_{n}$
+\end_inset
+
+ implica la de 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $l=+\infty$
+\end_inset
+
+ entonces la convergencia de 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ implica la de 
+\begin_inset Formula $\sum_{n}b_{n}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+
+\series bold
+Criterio de la raíz:
+\series default
+ Dada 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n}>0$
+\end_inset
+
+ y 
+\begin_inset Formula $a:=\lim_{n}\sqrt[n]{a_{n}}\in\mathbb{R}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a<1$
+\end_inset
+
+, la serie converge.
+ [...]
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a>1$
+\end_inset
+
+, la serie diverge.
+ [...]
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a=1$
+\end_inset
+
+ no se puede afirmar nada.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio del cociente:
+\series default
+ Sea 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n}>0$
+\end_inset
+
+ y 
+\begin_inset Formula $a:=\lim_{n}\frac{a_{n+1}}{a_{n}}\in\mathbb{R}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a<1$
+\end_inset
+
+, la serie converge.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a>1$
+\end_inset
+
+, la serie diverge.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio de condensación:
+\series default
+ Dada una sucesión 
+\begin_inset Formula $(a_{n})_{n}$
+\end_inset
+
+ monótona decreciente con 
+\begin_inset Formula $a_{n}>0$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\[
+\sum_{n=1}^{\infty}a_{n}\in\mathbb{R}\iff\sum_{n=1}^{\infty}2^{n}a_{2^{n}}\in\mathbb{R}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una serie 
+\begin_inset Formula $\sum_{n}a_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{n}\in\mathbb{R}$
+\end_inset
+
+ es 
+\series bold
+absolutamente convergente
+\series default
+ si 
+\begin_inset Formula $\sum_{n}|a_{n}|$
+\end_inset
+
+ es convergente.
+ Toda serie absolutamente convergente es convergente.
+ [...]
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+serie geométrica
+\series default
+ 
+\begin_inset Formula $\sum_{n=0}^{\infty}r^{n}$
+\end_inset
+
+ es convergente si 
+\begin_inset Formula $|r|<1$
+\end_inset
+
+ con suma 
+\begin_inset Formula $\frac{1}{1-r}$
+\end_inset
+
+ y divergente si 
+\begin_inset Formula $|r|\geq1$
+\end_inset
+
+.
+ La 
+\series bold
+serie armónica
+\series default
+ 
+\begin_inset Formula $\sum_{n=1}^{\infty}\frac{1}{n^{k}}$
+\end_inset
+
+ es convergente si 
+\begin_inset Formula $k>1$
+\end_inset
+
+ y divergente si 
+\begin_inset Formula $k\leq1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Completitud
+\end_layout
+
+\begin_layout Standard
+Una sucesión 
+\begin_inset Formula $(x_{n})$
+\end_inset
+
+ en un espacio métrico 
+\begin_inset Formula $(E,d)$
+\end_inset
+
+ es 
+\series bold
+de Cauchy
+\series default
+ si 
+\begin_inset Formula $\forall\varepsilon>0,\exists n_{\varepsilon}\in\mathbb{N}:\forall n,m\geq n_{\varepsilon},d(x_{n},x_{m})<\varepsilon$
+\end_inset
+
+.
+ Un espacio métrico es 
+\series bold
+completo
+\series default
+ si toda sucesión de Cauchy es convergente.
+ Un 
+\series bold
+espacio de Banach
+\series default
+ es un espacio normado completo.
+ Dadas dos normas 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert\cdot\Vert'$
+\end_inset
+
+ equivalentes sobre 
+\begin_inset Formula $E$
+\end_inset
+
+, 
+\begin_inset Formula $(E,\Vert\cdot\Vert)$
+\end_inset
+
+ es completo si y sólo si lo es 
+\begin_inset Formula $(E,\Vert\cdot\Vert')$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Existen 
+\begin_inset Formula $\alpha,\beta>0$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\alpha\Vert x\Vert\leq\Vert x\Vert'\leq\beta\Vert x\Vert$
+\end_inset
+
+, luego si 
+\begin_inset Formula $(x_{n})$
+\end_inset
+
+ es de Cauchy en 
+\begin_inset Formula $(E,\Vert\cdot\Vert)$
+\end_inset
+
+ también lo es en 
+\begin_inset Formula $(E,\Vert\cdot\Vert')$
+\end_inset
+
+ sin más que tomar 
+\begin_inset Formula $n_{\frac{\varepsilon}{\beta}}$
+\end_inset
+
+, y viceversa, y como 
+\begin_inset Formula $(x_{n})$
+\end_inset
+
+ converge a 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $(E,\Vert\cdot\Vert)$
+\end_inset
+
+ si y sólo si converge en 
+\begin_inset Formula $(E,\Vert\cdot\Vert')$
+\end_inset
+
+, la completitud de uno implica la del otro.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ es un espacio de Banach con cualquier norma.
+ 
+\series bold
+Demostración:
+\series default
+ Basta probar que 
+\begin_inset Formula $(\mathbb{R}^{n},\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ es completo.
+ Si 
+\begin_inset Formula $(x_{m})$
+\end_inset
+
+ es de Cauchy en 
+\begin_inset Formula $(\mathbb{R}^{n},\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+, como 
+\begin_inset Formula $|x_{mi}-x_{ki}|\leq\Vert x_{m}-x_{k}\Vert_{\infty}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $i\in\{1,\dots,n\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $(x_{mi})_{m}$
+\end_inset
+
+ es de Cauchy en 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ y por tanto convergente a un 
+\begin_inset Formula $x_{0i}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $(x_{m})_{m}$
+\end_inset
+
+ converge a 
+\begin_inset Formula $(x_{01},\dots,x_{0n})$
+\end_inset
+
+, y se tiene que 
+\begin_inset Formula $(\mathbb{R}^{n},\Vert\cdot\Vert)$
+\end_inset
+
+ es completo.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, 
+\begin_inset Formula $({\cal C}[a,b],\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+ es un espacio de Banach.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $(f_{n})_{n}$
+\end_inset
+
+ una sucesión de Cauchy en 
+\begin_inset Formula $({\cal C}[a,b],\Vert\cdot\Vert_{\infty})$
+\end_inset
+
+, fijado un 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe un 
+\begin_inset Formula $n_{0}$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $n,m\geq n_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $|f_{n}(x)-f_{m}(x)|<\frac{\varepsilon}{2}$
+\end_inset
+
+.
+ Por tanto para cada 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $(f_{n}(x))_{n}$
+\end_inset
+
+ es de Cauchy en 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ y converge pues a un valor 
+\begin_inset Formula $f(x)$
+\end_inset
+
+.
+ Ahora bien, dado un 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+, por la convergencia puntual que acabamos de probar existe 
+\begin_inset Formula $n_{x}\in\mathbb{N}$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $n\geq n_{x}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $|f_{n}(x)-f(x)|<\frac{\varepsilon}{2}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n>n_{0}$
+\end_inset
+
+, 
+\begin_inset Formula $|f_{n}(x)-f(x)|\leq|f_{n}(x)-f_{\max\{n_{0},n_{x}\}}(x)|+|f_{\max\{n_{0},n_{x}\}}(x)-f(x)|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon$
+\end_inset
+
+, y como 
+\begin_inset Formula $n_{0}$
+\end_inset
+
+ no depende de 
+\begin_inset Formula $x$
+\end_inset
+
+ (sólo de 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+), queda probada la convergencia uniforme.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ una sucesión en el espacio de Banach 
+\begin_inset Formula $(E,\Vert\cdot\Vert)$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{n}\Vert x_{n}\Vert<+\infty$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sum_{n}x_{n}$
+\end_inset
+
+ converge.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Tenemos que 
+\begin_inset Formula $\sum_{n}\Vert x_{n}\Vert$
+\end_inset
+
+, por ser convergente, es de Cachy, por lo que dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe 
+\begin_inset Formula $n_{\varepsilon}$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $n,m>n_{\varepsilon}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $|\sum_{k=1}^{n}\Vert x_{k}\Vert-\sum_{k=1}^{m}\Vert x_{k}\Vert|<\varepsilon$
+\end_inset
+
+.
+ Pero por la desigualdad triangular 
+\begin_inset Formula $\Vert\sum_{k=1}^{n}x_{k}-\sum_{k=1}^{m}x_{k}\Vert\leq|\sum_{k=1}^{n}\Vert x_{k}\Vert-\sum_{k=1}^{m}\Vert x_{k}\Vert|<\varepsilon$
+\end_inset
+
+, luego 
+\begin_inset Formula $\sum_{n}x_{n}$
+\end_inset
+
+ también es de Cauchy y por tanto convergente por estar en un espacio completo.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv1/n2.lyx b/fvv1/n2.lyx
new file mode 100644
index 0000000..c190e9d
--- /dev/null
+++ b/fvv1/n2.lyx
@@ -0,0 +1,1010 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Derivadas
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $E$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+ normados, 
+\begin_inset Formula $\Omega\subseteq E$
+\end_inset
+
+ abierto y 
+\begin_inset Formula $f:\Omega\rightarrow F$
+\end_inset
+
+, dados 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $u\in E$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+derivable
+\series default
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ según 
+\begin_inset Formula $u$
+\end_inset
+
+ si existe la 
+\series bold
+derivada direccional
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $x$
+\end_inset
+
+ según 
+\begin_inset Formula $u$
+\end_inset
+
+, dada por
+\begin_inset Formula 
+\[
+d_{u}f(x_{0}):=\lim_{t\rightarrow0}\frac{f(a+tu)-f(a)}{t}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $u=\vec{e}_{i}$
+\end_inset
+
+ es el vector 
+\begin_inset Formula $i$
+\end_inset
+
+-ésimo de la base canónica hablamos de la 
+\series bold
+derivada parcial
+\series default
+ 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima, que denotamos
+\begin_inset Formula 
+\[
+\frac{\partial f}{\partial x_{i}}(a):=d_{i}f(a):=d_{\vec{e}_{i}}f(a)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\vec{u}=\lambda\vec{v}$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $d_{\vec{u}}f(t)$
+\end_inset
+
+ existe si y sólo si existe 
+\begin_inset Formula $d_{\vec{v}}f(t)$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues 
+\begin_inset Formula 
+\[
+d_{\vec{u}}f(t)=\lim_{t\rightarrow0}\frac{f(a+t\lambda\vec{v})-f(a)}{t}=\lim_{t\rightarrow0}\frac{f(a+t\lambda\vec{v})-f(a)}{t\lambda}\lambda\overset{t\lambda\rightarrow0}{=}\lambda d_{\vec{v}}f(t)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:(a,b)\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es derivable en 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ si cada una de sus coordenadas lo es, y entonces 
+\begin_inset Formula $f'(x_{0})=(f'_{1}(x_{0}),\dots,f'_{n}(x_{0}))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Diferenciales
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\Omega\subseteq E$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $f:\Omega\rightarrow F$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+diferenciable
+\series default
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ si existe una aplicación lineal 
+\begin_inset Formula $L:E\rightarrow F$
+\end_inset
+
+ tal que
+\begin_inset Formula 
+\[
+\lim_{h\rightarrow0}\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Esta aplicación es la 
+\series bold
+diferencial
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+, denotada por 
+\begin_inset Formula $df(a)$
+\end_inset
+
+, y si existe es única.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $L,M:E\rightarrow F$
+\end_inset
+
+ dos diferenciales de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{L(h)-M(h)}{\Vert h\Vert}=\lim_{h\rightarrow0}\left(\frac{f(a+h)-f(a)-M(h)}{\Vert h\Vert}-\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}\right)=0-0=0$
+\end_inset
+
+, pero entonces, dado 
+\begin_inset Formula $v\neq0$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $0=\lim_{t\rightarrow0^{+}}\frac{L(tv)-M(tv)}{\Vert tv\Vert}=\lim_{t\rightarrow0^{+}}\frac{L(v)-M(v)}{\Vert v\Vert}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $L(v)=M(v)$
+\end_inset
+
+ y 
+\begin_inset Formula $L=M$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Escribimos 
+\begin_inset Formula $L\equiv M$
+\end_inset
+
+ si 
+\begin_inset Formula $M$
+\end_inset
+
+ es la matriz asociada a la aplicación lineal 
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ con diferencial 
+\begin_inset Formula $L$
+\end_inset
+
+ si y sólo si cada 
+\begin_inset Formula $f_{i}:\mathbb{R}^{m}\rightarrow\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $i\in\{1,\dots,n\}$
+\end_inset
+
+, es diferenciable con diferencial 
+\begin_inset Formula $L_{i}$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues 
+\begin_inset Formula $\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}\rightarrow0\iff\forall i\in\{1,\dots,n\},\left(\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}\right)_{i}\rightarrow0\iff\frac{f_{i}(a+h)-f_{i}(a)-L_{i}(h)}{\Vert h\Vert}\rightarrow0$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:\Omega\subseteq E\rightarrow F$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+, también es continua en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $a$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\lim_{x\rightarrow a}f(x)=f(a)$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\lim_{h\rightarrow0}f(a+h)-f(a)=0$
+\end_inset
+
+.
+ Pero si 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}=0$
+\end_inset
+
+, multiplicando por 
+\begin_inset Formula $\Vert h\Vert$
+\end_inset
+
+, que tiende a 0, tenemos 
+\begin_inset Formula $\lim_{h\rightarrow0}f(a+h)-f(a)-L(h)=0$
+\end_inset
+
+, y como 
+\begin_inset Formula $L(h)$
+\end_inset
+
+ tiende a 0 nos queda la expresión de arriba.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ (
+\begin_inset Formula $\Omega$
+\end_inset
+
+ abierto) si para todo 
+\begin_inset Formula $P\in\Omega$
+\end_inset
+
+ existen todas las derivadas parciales 
+\begin_inset Formula $\frac{\partial f_{i}}{\partial x_{j}}(P)$
+\end_inset
+
+ y son continuas en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Podemos suponer 
+\begin_inset Formula $n=1$
+\end_inset
+
+, pues de lo contrario basta probar que cada 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Se trata pues de probar que 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{f(a+h)-f(a)-\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}(a)h_{i}}{\Vert h\Vert}=0$
+\end_inset
+
+, lo que ocurre si y sólo si
+\begin_inset Formula 
+\begin{eqnarray*}
+0 & = & \lim_{h\rightarrow0}\frac{|f(a+h)-f(a)-\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}(a)h_{i}|}{\Vert h\Vert}=\lim_{h\rightarrow0}\frac{f(a+\sum_{i=1}^{m}h_{i}\vec{e}_{i})-f(a)-\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}(a)h_{i}}{\Vert h\Vert}\\
+ & = & \lim_{h\rightarrow0}\frac{\left|\sum_{i=1}^{m}\left(f(a+h_{1}\vec{e}_{1}+\dots+h_{i}\vec{e}_{i})-f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1})-\frac{\partial f}{\partial x_{i}}(a)h_{i}\right)\right|}{\Vert h\Vert}
+\end{eqnarray*}
+
+\end_inset
+
+El último sumatorio con sus dos primeros elementos forma una 
+\series bold
+suma telescópica
+\series default
+: todos los elementos se anulan salvo el primero y el último.
+ Sabemos que cada 
+\begin_inset Formula $a+h_{1}\vec{e}_{1}+\dots+h_{i}\vec{e}_{i}$
+\end_inset
+
+ está en el dominio de 
+\begin_inset Formula $f$
+\end_inset
+
+ porque 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es abierto y 
+\begin_inset Formula $h$
+\end_inset
+
+ se supone lo suficientemente pequeño.
+ Ahora llamamos 
+\begin_inset Formula $\varphi_{i}(t):=f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1}+t\vec{e}_{i})$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\varphi'_{i}(t)=\frac{\partial f}{\partial x_{i}}(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1}+t\vec{e}_{i})$
+\end_inset
+
+, y 
+\begin_inset Formula $\Delta_{i}:=\varphi_{i}(h_{i})-\varphi_{i}(0)=f(a+h_{1}\vec{e}_{1}+\dots+h_{i}\vec{e}_{i})-f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1})=\varphi'_{i}(\xi_{i})h_{i}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $\xi_{i}$
+\end_inset
+
+ entre 0 y 
+\begin_inset Formula $h_{i}$
+\end_inset
+
+, que tiende a 0.
+ Sustituyendo nos queda que lo anterior es igual a 
+\begin_inset Formula 
+\[
+\lim_{h\rightarrow0}\frac{\left|\sum_{i=1}^{m}\varphi'_{i}(\xi_{i})h_{i}-\frac{\partial f}{\partial x_{i}}(a)h_{i}\right|}{\Vert h\Vert}
+\]
+
+\end_inset
+
+Entonces,
+\begin_inset Formula 
+\begin{eqnarray*}
+0 & \leq & \frac{\left|\sum_{i=1}^{m}\varphi'_{i}(\xi_{i})h_{i}-\frac{\partial f}{\partial x_{i}}(a)h_{i}\right|}{\Vert h\Vert_{\infty}}\\
+ & \leq & \left|\sum_{i=1}^{m}\frac{\partial f}{\partial x_{i}}(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1}+\xi_{i}\vec{e}_{i})-\frac{\partial f}{\partial x_{i}}(a)\right|\frac{\left|h_{i}\right|}{\Vert h\Vert_{\infty}}\rightarrow0
+\end{eqnarray*}
+
+\end_inset
+
+Que esta última expresión tienda a 0 se debe a que 
+\begin_inset Formula $0\leq\frac{|h_{i}|}{\Vert h\Vert_{\infty}}\leq1$
+\end_inset
+
+ y a que las derivadas parciales de 
+\begin_inset Formula $f$
+\end_inset
+
+ sean continuas y por tanto 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{\partial f}{\partial x_{i}}(a+\dots)=\frac{\partial f}{\partial x_{i}}(\lim_{h\rightarrow0}(a+\dots))$
+\end_inset
+
+.
+ Entonces, por la regla del sandwich, el límite inicial tiende a 0.
+ Hemos utilizado la norma 
+\begin_inset Formula $\Vert\cdot\Vert_{\infty}$
+\end_inset
+
+, pero como dada una norma 
+\begin_inset Formula $\Vert\cdot\Vert$
+\end_inset
+
+ 
+\begin_inset Formula $\exists\alpha,\beta>0:\forall h,\alpha\leq\frac{\Vert h\Vert_{\infty}}{\Vert h\Vert}\leq\beta$
+\end_inset
+
+, la convergencia a 0 no depende de la norma que tomemos.
+\end_layout
+
+\begin_layout Section
+Regla de la cadena
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+regla de la cadena
+\series default
+ afirma que si 
+\begin_inset Formula ${\cal U}\subseteq\mathbb{R}^{m}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal V}\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ son abiertos, 
+\begin_inset Formula $a\in{\cal U}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal U}\overset{f}{\rightarrow}{\cal V}\overset{g}{\rightarrow}\mathbb{R}^{k}$
+\end_inset
+
+ son diferenciables en 
+\begin_inset Formula $a$
+\end_inset
+
+ y en 
+\begin_inset Formula $f(a)$
+\end_inset
+
+, respectivamente, entonces 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $d(g\circ f)(a)=dg(f(a))\circ df(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $L:=df(a):\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $S:=dg(f(a)):\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$
+\end_inset
+
+, tenemos que 
+\begin_inset Formula 
+\[
+\lim_{h\rightarrow0}\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}=\lim_{\eta\rightarrow0}\frac{g(f(a)+\eta)-g(f(a))-S(\eta)}{\Vert\eta\Vert}=0
+\]
+
+\end_inset
+
+y queremos ver que 
+\begin_inset Formula 
+\[
+\lim_{h\rightarrow0}\frac{g(f(a+h))-g(f(a))-S(L(h))}{\Vert h\Vert}=0
+\]
+
+\end_inset
+
+Si llamamos 
+\begin_inset Formula $\eta:=f(a+h)-f(a)$
+\end_inset
+
+, que tiende a 0 por la continuidad de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\begin{multline*}
+\lim_{h\rightarrow0}\frac{g(f(a+h))-g(f(a))-S(L(h))}{\Vert h\Vert}=\lim_{h\rightarrow0}\frac{g(f(a)+\eta)-g(f(a))-S(\eta)}{\Vert h\Vert}+\frac{S(\eta)-S(L(h))}{\Vert h\Vert}\\
+=\lim_{h\rightarrow0}\frac{g(f(a)+\eta)-g(f(a))-S(\eta)}{\Vert\eta\Vert}\frac{\Vert\eta\Vert}{\Vert h\Vert}-S\left(\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}\right)
+\end{multline*}
+
+\end_inset
+
+Como 
+\begin_inset Formula $S\left(\frac{f(a+h)-f(a)-L(h)}{\Vert h\Vert}\right)\rightarrow0$
+\end_inset
+
+ usando la linealidad de 
+\begin_inset Formula $S$
+\end_inset
+
+ y su continuidad (que se deduce de su linealidad), y como 
+\begin_inset Formula $\frac{g(f(a)+\eta)-g(f(a))-S(\eta)}{\Vert\eta\Vert}\rightarrow0$
+\end_inset
+
+, el límite tenderá a 0 si y sólo si 
+\begin_inset Formula $\frac{\Vert\eta\Vert}{\Vert h\Vert}$
+\end_inset
+
+ es acotado, pero 
+\begin_inset Formula 
+\begin{eqnarray*}
+0 & \leq & \frac{\Vert\eta\Vert}{\Vert h\Vert}=\frac{\Vert f(a+h)-f(a)-L(h)+L(h)\Vert}{\Vert h\Vert}\\
+ & \leq & \frac{\Vert f(a+h)-f(a)-L(h)\Vert}{\Vert h\Vert}+\frac{\Vert L(h)\Vert}{\Vert h\Vert}\rightarrow0+\frac{\Vert L(h)\Vert}{\Vert h\Vert}\leq\frac{\Vert L\Vert\Vert h\Vert}{\Vert h\Vert}<+\infty
+\end{eqnarray*}
+
+\end_inset
+
+ por la continuidad de 
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Incremento finito
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema del incremento finito
+\series default
+ afirma que, sean 
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $a,b\in\Omega$
+\end_inset
+
+ con el segmento 
+\begin_inset Formula $[a,b]\subseteq\Omega$
+\end_inset
+
+ y 
+\begin_inset Formula $L:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ lineal, si 
+\begin_inset Formula $\Vert df(x)\Vert\leq M$
+\end_inset
+
+ para todo 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $\Vert f(b)-f(a)\Vert\leq M\Vert b-a\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Fijado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, sabemos que para 
+\begin_inset Formula $x\in[a,b]$
+\end_inset
+
+, 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{f(x+h)-f(x)-df(x)(h)}{\Vert h\Vert}=0$
+\end_inset
+
+ y por tanto existe 
+\begin_inset Formula $\delta_{x}>0$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $\Vert h\Vert<\delta_{x}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula 
+\[
+\Vert f(x+h)-f(x)-df(x)(h)\Vert<\varepsilon\Vert h\Vert
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula 
+\[
+\Vert f(x+h)-f(x)\Vert-\Vert df(x)(h)\Vert\leq\Vert f(x+h)-f(x)-df(x)(h)\Vert<\varepsilon\Vert h\Vert
+\]
+
+\end_inset
+
+ y por tanto
+\begin_inset Formula 
+\[
+\Vert f(x+h)-f(x)\Vert<\varepsilon\Vert h\Vert+\Vert df(x)(h)\Vert\leq\varepsilon\Vert h\Vert+\Vert df(x)\Vert\Vert h\Vert\leq(\varepsilon+M)\Vert h\Vert
+\]
+
+\end_inset
+
+Esta desigualdad depende de 
+\begin_inset Formula $\delta_{x}$
+\end_inset
+
+ y por tanto de 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $\{B(x,\frac{\delta_{x}}{2})\}_{x\in[a,b]}$
+\end_inset
+
+ un re
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+cu
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+bri
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+mien
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+to por abiertos de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\{B_{i}\}_{i=1}^{k}:=\{B(x_{i},\frac{\delta_{x_{i}}}{2})\}_{i=1}^{k}$
+\end_inset
+
+ un subrecubrimiento finito del que suponemos que no podemos quitar ninguna
+ bola.
+ Ahora llamamos 
+\begin_inset Formula $x_{0}:=a$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{k+1}:=b$
+\end_inset
+
+ y suponemos 
+\begin_inset Formula $a=x_{0}<x_{1}<\dots<x_{k}<x_{k+1}=b$
+\end_inset
+
+.
+ Por la desigualdad anterior, para 
+\begin_inset Formula $x,y\in[a,b]$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert y-x\Vert<\delta_{x}$
+\end_inset
+
+ o 
+\begin_inset Formula $\Vert x-y\Vert<\delta_{y}$
+\end_inset
+
+, 
+\begin_inset Formula $\Vert f(y)-f(x)\Vert\leq(M+\varepsilon)\Vert x-y\Vert$
+\end_inset
+
+.
+ El segmento 
+\begin_inset Formula $[x_{i},x_{i+1}]$
+\end_inset
+
+ queda cubierto por 
+\begin_inset Formula $B_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $B_{i+1}$
+\end_inset
+
+, pues si hiciera falta además 
+\begin_inset Formula $B_{j}$
+\end_inset
+
+ con 
+\begin_inset Formula $j\neq i,i+1$
+\end_inset
+
+ para cubrirlo sería 
+\begin_inset Formula $x_{j}<x_{i}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $B_{i}\subseteq B_{j}$
+\end_inset
+
+ o 
+\begin_inset Formula $x_{j}>x_{i+1}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $B_{i+1}\subseteq B_{j}$
+\end_inset
+
+, pero entonces podríamos quitar una bola del recubrimiento
+\begin_inset Formula $\#$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\Vert x_{i+1}-x_{i}\Vert<\frac{\delta_{x_{i}}}{2}+\frac{\delta_{x_{i+1}}}{2}\leq\max\{\delta_{x_{i}},\delta_{x_{i+1}}\}$
+\end_inset
+
+.
+ Finalmente tenemos que 
+\begin_inset Formula $\Vert f(b)-f(a)\Vert=\Vert f(x_{k+1})-f(x_{k})+\dots+f(x_{1})-f(x_{0})\Vert\leq\sum_{i=0}^{k}\Vert f(x_{i+1})-f(x_{i})\Vert\leq\sum_{i=0}^{k}\Vert x_{i+1}-x_{i}\Vert(M+\varepsilon)$
+\end_inset
+
+ y, como todos los 
+\begin_inset Formula $x_{i+1}-x_{i}$
+\end_inset
+
+ tienen la forma 
+\begin_inset Formula $\lambda(b-a)$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda>0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sum_{i=0}^{k}\Vert x_{i+1}-x_{i}\Vert(M+\varepsilon)=\Vert b-a\Vert(M+\varepsilon)$
+\end_inset
+
+.
+ Como esto se da para todo 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, el resultado queda probado.
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv1/n3.lyx b/fvv1/n3.lyx
new file mode 100644
index 0000000..776351a
--- /dev/null
+++ b/fvv1/n3.lyx
@@ -0,0 +1,1046 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es 
+\series bold
+dos veces diferenciable
+\series default
+ o 
+\series bold
+de clase
+\series default
+ 
+\begin_inset Formula ${\cal C}^{2}$
+\end_inset
+
+ en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ si 
+\begin_inset Formula $f$
+\end_inset
+
+ es diferenciable en 
+\begin_inset Formula ${\cal U}\in{\cal E}(a)$
+\end_inset
+
+ y 
+\begin_inset Formula $df:{\cal U}\rightarrow{\cal L}(\mathbb{R}^{m},\mathbb{R}^{n})\equiv M_{n\times m}(\mathbb{R})\equiv\mathbb{R}^{nm}$
+\end_inset
+
+ (la aplicación que a cada elemento de 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ le asigna un vector en 
+\begin_inset Formula $\mathbb{R}^{nm}$
+\end_inset
+
+ que contiene, en algún orden, los elementos de la matriz asociada a la
+ diferencial del elemento) es diferenciable en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Por inducción se define el ser 
+\begin_inset Formula $n$
+\end_inset
+
+ veces diferenciable o de clase 
+\begin_inset Formula ${\cal C}^{n}$
+\end_inset
+
+, y el ser infinitamente diferenciable o de clase 
+\begin_inset Formula ${\cal C}^{\infty}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Denotamos la derivada parcial 
+\begin_inset Formula $k$
+\end_inset
+
+-ésima de la derivada parcial 
+\begin_inset Formula $j$
+\end_inset
+
+-ésima de la 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima coordenada de 
+\begin_inset Formula $f$
+\end_inset
+
+, o la 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima coordenada de la doble derivada parcial respecto a 
+\begin_inset Formula $x_{j}$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{k}$
+\end_inset
+
+, como 
+\begin_inset Formula $\frac{\partial^{2}f_{i}}{\partial x_{j}\partial x_{k}}$
+\end_inset
+
+, y si 
+\begin_inset Formula $j=k$
+\end_inset
+
+, también escribimos 
+\begin_inset Formula $\frac{\partial^{2}f_{i}}{\partial x_{j}^{2}}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene derivadas parciales segundas 
+\begin_inset Formula $\frac{\partial f_{i}}{\partial x_{k}\partial x_{j}}$
+\end_inset
+
+ todas continuas en 
+\begin_inset Formula $a$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es dos veces diferenciable en 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Matriz hessiana
+\end_layout
+
+\begin_layout Standard
+Del mismo modo que podemos pensar en la diferencial de una función diferenciable
+ como 
+\begin_inset Formula $df(a):\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\vec{u}\mapsto d_{\vec{u}}f(a)$
+\end_inset
+
+, llamamos 
+\series bold
+diferencial segunda
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ a la aplicación 
+\begin_inset Formula $d^{2}f(a):\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $(\vec{u},\vec{v})\mapsto d_{\vec{v}}d_{\vec{u}}f(a)$
+\end_inset
+
+, y vemos que esta es una aplicación bilineal.
+\end_layout
+
+\begin_layout Standard
+La matriz de 
+\begin_inset Formula $d^{2}f(a):\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$
+\end_inset
+
+, dada por
+\begin_inset Formula 
+\[
+\left(\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\right)_{ij}
+\]
+
+\end_inset
+
+ se denomina 
+\series bold
+matriz hessiana
+\series default
+.
+ Así, si 
+\begin_inset Formula $M$
+\end_inset
+
+ es la matriz hessiana de 
+\begin_inset Formula $f$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+d^{2}f(a)(\vec{u},\vec{v})=\left(\begin{array}{ccc}
+- & \vec{u} & -\end{array}\right)M\left(\begin{array}{c}
+|\\
+\vec{v}\\
+|
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sea 
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{2}\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $a=(x_{0},y_{0})\in\Omega$
+\end_inset
+
+, si 
+\begin_inset Formula $\frac{\partial^{2}f}{\partial x\partial y}$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{\partial^{2}f}{\partial y\partial x}$
+\end_inset
+
+ están definidas en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ y son continuas en 
+\begin_inset Formula $a$
+\end_inset
+
+, entonces su valor en 
+\begin_inset Formula $a$
+\end_inset
+
+ coincide.
+ Esto significa que la matriz Hessiana es simétrica.
+ 
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ es abierto, existe 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ tal que 
+\begin_inset Formula $B_{\infty}(a,\varepsilon)=(x_{0}-\varepsilon,x_{0}+\varepsilon)\times(y_{0}-\varepsilon,y_{0}+\varepsilon)\subseteq\Omega$
+\end_inset
+
+.
+ Fijamos 
+\begin_inset Formula $t\in(x_{0}-\varepsilon,y_{0}+\varepsilon)$
+\end_inset
+
+ y 
+\begin_inset Formula $s\in(y_{0}-\varepsilon,y_{0}+\varepsilon)$
+\end_inset
+
+, y consideramos 
+\begin_inset Formula $\Delta_{t,s}:=f(t,s)-f(t,y_{0})-f(x_{0},s)+f(x_{0},y_{0})$
+\end_inset
+
+.
+ Si ahora llamamos 
+\begin_inset Formula $F_{\overline{s}}(\overline{t}):=f(\overline{t},\overline{s})-f(\overline{t},y_{0})$
+\end_inset
+
+, vemos que 
+\begin_inset Formula $F_{\overline{s}}(\overline{t})$
+\end_inset
+
+ es derivable con 
+\begin_inset Formula $F'_{\overline{s}}(\overline{t})=\frac{\partial f}{\partial x}(\overline{t},\overline{s})-\frac{\partial f}{\partial x}(\overline{t},y_{0})$
+\end_inset
+
+ y que entonces 
+\begin_inset Formula $\Delta_{t,s}=F_{s}(t)-F_{s}(x_{0})=F'_{\overline{s}}(\xi_{t,s})(t-x_{0})=\left(\frac{\partial f}{\partial x}(\xi_{t,s},s)-\frac{\partial f}{\partial x}(\xi_{t,s},y_{0})\right)(t-x_{0})\overset{\Phi(\overline{s}):=\frac{\partial f}{\partial x}(\xi_{t,s},\overline{s})}{=}(\Phi(s)-\Phi(y_{0}))(t-x_{0})\overset{\Phi\text{ derivable por hipótesis}}{=}\Phi'(\eta_{t,s})(s-y_{0})(t-x_{0})=\frac{\partial^{2}f}{\partial x\partial y}(\xi_{t,s},\eta_{t,s})(s-y_{0})(t-x_{0})$
+\end_inset
+
+.
+ Permutando los papeles de las dos coordenadas (definiendo 
+\begin_inset Formula $\sigma_{\overline{t}}(\overline{s}):=f(\overline{t},\overline{s})-f(x,\overline{s})$
+\end_inset
+
+) obtenemos que 
+\begin_inset Formula $\Delta_{t,s}=\sigma_{\overline{t}}(\overline{s})-\sigma_{\overline{t}}(y_{0})=\frac{\partial^{2}f}{\partial y\partial x}(\hat{\xi}_{t,s},\hat{\eta}_{t,s})(s-y_{0})(t-x_{0})$
+\end_inset
+
+.
+ Haciendo ahora tender 
+\begin_inset Formula $(t,s)$
+\end_inset
+
+ a 
+\begin_inset Formula $(x_{0},y_{0})$
+\end_inset
+
+, por la regla del sandwich 
+\begin_inset Formula $(\xi_{t,s},\eta_{ts})$
+\end_inset
+
+ y 
+\begin_inset Formula $(\hat{\xi}_{t,s},\hat{\eta}_{t,s})$
+\end_inset
+
+ también tienden a 
+\begin_inset Formula $(x_{0},y_{0})$
+\end_inset
+
+, y aplicando la continuidad de las derivadas parciales dobles en 
+\begin_inset Formula $a$
+\end_inset
+
+, nos queda finalmente que 
+\begin_inset Formula $\frac{\partial^{2}f}{\partial x\partial y}(x_{0},y_{0})=\frac{\partial^{2}f}{\partial y\partial x}(x_{0},y_{0})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Desarrollos de Taylor
+\end_layout
+
+\begin_layout Standard
+Despejando de la definición de diferencial, nos queda que 
+\begin_inset Formula $f(a+h)=f(a)+df(a)(h)+o(\Vert h\Vert)$
+\end_inset
+
+, lo que podemos interpretar como una aproximación de 
+\begin_inset Formula $f(x)$
+\end_inset
+
+ cerca de 
+\begin_inset Formula $a$
+\end_inset
+
+ por un polinomio de grado 1.
+ Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}$
+\end_inset
+
+ es dos veces diferenciable en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f(a+h)=f(a)+df(a)(h)+\frac{1}{2}d^{2}f(a)(h,h)+o(\Vert h\Vert^{2})$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $R(h):=f(a+h)-f(a)-df(a)(h)-\frac{1}{2}d^{2}f(a)(h,h)$
+\end_inset
+
+, y hemos de ver que 
+\begin_inset Formula $\lim_{h\rightarrow0}\frac{R(h)}{\Vert h\Vert^{2}}=0$
+\end_inset
+
+.
+ Como todas las normas en 
+\begin_inset Formula $\mathbb{R}^{m}$
+\end_inset
+
+ son equivalentes, elegimos 
+\begin_inset Formula $\Vert\cdot\Vert_{\infty}$
+\end_inset
+
+.
+ Usamos el teorema del incremento finito, que afirma que si 
+\begin_inset Formula $R$
+\end_inset
+
+ es diferenciable y 
+\begin_inset Formula $\Vert dR(\xi)\Vert\leq M\forall\xi\in[0,h]$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\Vert R(h)-R(0)\Vert\leq M\cdot\Vert h-0\Vert$
+\end_inset
+
+.
+ 
+\begin_inset Formula $R$
+\end_inset
+
+ es diferenciable al ser la suma de 
+\begin_inset Formula $f(c+h)$
+\end_inset
+
+ y un polinomio de grado máximo 2.
+ Para estimar 
+\begin_inset Formula $\Vert dR\Vert$
+\end_inset
+
+ vemos que 
+\begin_inset Formula $R(a)=f(a+h)-f(a)-\sum_{i}\frac{\partial f}{\partial x_{i}}(a)h_{i}-\frac{1}{2}\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)h_{i}h_{j}$
+\end_inset
+
+, y usando la 
+\begin_inset Formula $\delta$
+\end_inset
+
+ de Kronecker,
+\begin_inset Formula 
+\[
+\frac{\partial}{\partial x_{k}}\frac{1}{2}\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)h_{i}h_{j}=\frac{1}{2}\sum_{i,j}\left(\frac{\partial f}{\partial x_{i}x_{j}}(a)\delta_{ik}+\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)\delta_{jk}\right)=\frac{1}{2}\cdot2d\left(\frac{\partial f}{\partial x_{k}}\right)(a)(h)
+\]
+
+\end_inset
+
+ Por tanto
+\begin_inset Formula 
+\[
+\frac{\partial R}{\partial x_{k}}=\frac{\partial f}{\partial x_{k}}(a+h)-0-\frac{\partial f}{\partial x_{k}}(a)-d\left(\frac{\partial f}{\partial x_{k}}\right)(a)(h)=:\psi_{k}(h)\Vert h\Vert
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $\lim_{h\rightarrow0}\psi(h)=0$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\frac{\partial R}{\partial x_{k}}$
+\end_inset
+
+ es continua, definiendo el compacto 
+\begin_inset Formula $[0,h]$
+\end_inset
+
+ como 
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+
+\begin_inset Formula $\{th\}_{t\in[0,1]}$
+\end_inset
+
+
+\family default
+\series default
+\shape default
+\size default
+\emph default
+\bar default
+\strikeout default
+\xout default
+\uuline default
+\uwave default
+\noun default
+\color inherit
+ existe un punto 
+\begin_inset Formula $t_{k,h}h\in[0,h]$
+\end_inset
+
+ tal que 
+\begin_inset Formula 
+\[
+\frac{\partial R}{\partial x_{k}}(t_{k,h}h)=\max\left\{ \frac{\partial R}{\partial x_{k}}(\xi)\right\} _{\xi\in[0,h]}
+\]
+
+\end_inset
+
+Por esto, y como 
+\begin_inset Formula $dR(\xi)\equiv\left(\frac{\partial R}{\partial x_{1}}(\xi),\dots,\frac{\partial R}{\partial x_{m}}(\xi)\right)$
+\end_inset
+
+, existe 
+\begin_inset Formula $C$
+\end_inset
+
+ tal que 
+\begin_inset Formula 
+\[
+\Vert dR(\xi)\Vert\leq C\cdot\Vert dR(a)\Vert_{\infty}=C\cdot\max\left\{ \left|\frac{\partial R}{\partial x_{k}}(a)\right|\right\} _{k=1}^{n}
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $\xi\in[0,h]$
+\end_inset
+
+, y por el teorema del incremento finito, si 
+\begin_inset Formula $p$
+\end_inset
+
+ es tal que 
+\begin_inset Formula 
+\[
+\left|\frac{\partial R}{\partial x_{p}}(a)\right|=\max\left\{ \left|\frac{\partial R}{\partial x_{k}}(a)\right|\right\} _{k=1}^{n}
+\]
+
+\end_inset
+
+tenemos 
+\begin_inset Formula 
+\[
+\Vert R(h)\Vert=\Vert R(h)-R(0)\Vert\leq C\left|\frac{\partial R}{\partial x_{p}}(t_{p,h})\right|\Vert h\Vert=C\psi_{p}(t_{p,h}h)\Vert h\Vert\Vert h\Vert=C\psi_{p}(t_{p,h}h)\Vert h\Vert^{2}
+\]
+
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\frac{|R(h)|}{\Vert h\Vert^{2}}\leq C\psi_{p}(t_{p,h}h)\rightarrow0$
+\end_inset
+
+, lo que prueba el teorema.
+\end_layout
+
+\begin_layout Section
+Extremos relativos
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $V$
+\end_inset
+
+ es un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial con 
+\begin_inset Formula $k:=\dim_{K}(V)<+\infty$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma:V\times V\rightarrow\mathbb{R}$
+\end_inset
+
+ una aplicación bilineal, existe 
+\begin_inset Formula $A=(a_{ij})\in{\cal M}_{k}(K)$
+\end_inset
+
+ asociado a 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ y podemos definir
+\begin_inset Formula 
+\[
+\Delta_{1}=\left|a_{11}\right|,\Delta_{2}=\left|\begin{array}{cc}
+a_{11} & a_{12}\\
+a_{21} & a_{22}
+\end{array}\right|,\dots,\Delta_{k}=\left|\begin{array}{ccc}
+a_{11} & \cdots & a_{1k}\\
+\vdots & \ddots & \vdots\\
+a_{k1} & \cdots & a_{kk}
+\end{array}\right|
+\]
+
+\end_inset
+
+Entonces un 
+\series bold
+teorema
+\series default
+ de álgebra nos dice que 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ es:
+\end_layout
+
+\begin_layout Enumerate
+Semidefinida positiva si y sólo si 
+\begin_inset Formula $\Delta_{i}\geq0\forall i$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Semidefinida negativa si y sólo si 
+\begin_inset Formula $\Delta_{i}(-1)^{i}\geq0\forall i$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Definida positiva si y sólo si 
+\begin_inset Formula $\Delta_{i}>0\forall i$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Definida negativa si y sólo si 
+\begin_inset Formula $\Delta_{i}(-1)^{i}>0\forall i$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sea 
+\begin_inset Formula $\Omega\subseteq\mathbb{R}^{m}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ alcanza en 
+\begin_inset Formula $a$
+\end_inset
+
+ un extremo relativo entonces 
+\begin_inset Formula $df(a)=0$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Podemos suponer que alcanza un máximo.
+ Entonces 
+\begin_inset Formula $\exists{\cal U}\in{\cal E}(a):f(x)\leq f(a)\forall x\in{\cal U}$
+\end_inset
+
+, luego si para 
+\begin_inset Formula $i\in\{1,\dots,m\}$
+\end_inset
+
+ definimos 
+\begin_inset Formula $\varphi_{i}(t):=f(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})$
+\end_inset
+
+, fijado 
+\begin_inset Formula $i$
+\end_inset
+
+, 
+\begin_inset Formula $\exists\varepsilon>0:\forall t\in(a-\varepsilon,a+\varepsilon),\varphi_{i}(t)\leq\varphi_{i}(a_{i})$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{\partial f}{\partial x_{i}}(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})=\varphi'(t)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\frac{\partial f}{\partial x_{i}}(a)=\varphi'(a_{i})=0$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+df(a)=\left(\begin{array}{ccc}
+\frac{\partial f}{\partial x_{1}}(a) & \cdots & \frac{\partial f}{\partial x_{m}}(a)\end{array}\right)=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es de clase 
+\begin_inset Formula ${\cal C}^{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $df(a)=0$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\begin{eqnarray*}
+d^{2}f(a)\text{ definida positiva} & \implies & f\text{ tiene un mínimo estricto en }a\implies\\
+\implies f\text{ tiene un mínimo en }a & \implies & d^{2}f(a)\text{ semidefinida positiva}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Consideremos el desarrollo de Taylor de 
+\begin_inset Formula $f$
+\end_inset
+
+ de orden 2 en 
+\begin_inset Formula $a$
+\end_inset
+
+, que como 
+\begin_inset Formula $df(a)=0$
+\end_inset
+
+, queda como
+\begin_inset Formula 
+\begin{eqnarray*}
+f(x) & = & f(a)+\frac{1}{2}d^{2}f(a)(x-a,x-a)+o(\Vert x-a\Vert^{2})\\
+ & = & f(a)+\frac{1}{2}\Vert x-a\Vert^{2}d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+o(\Vert x-a\Vert^{2})\\
+ & = & f(a)+\frac{1}{2}\Vert x-a\Vert^{2}\left(d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}\right)
+\end{eqnarray*}
+
+\end_inset
+
+suponiendo 
+\begin_inset Formula $x\neq a$
+\end_inset
+
+.
+ Pero 
+\begin_inset Formula $\frac{x-a}{\Vert x-a\Vert}\in\{y\in\mathbb{R}^{m}:\Vert y\Vert=1\}=:K$
+\end_inset
+
+, que es compacto por ser cerrado y acotado, y 
+\begin_inset Formula $\Phi:\mathbb{R}^{m}\rightarrow\mathbb{R}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\Phi(u):=d^{2}f(a)(u,u)=\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)u_{i}u_{j}$
+\end_inset
+
+ es continua, luego 
+\begin_inset Formula $\Phi(K)=\left\{ d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)\right\} _{x\in\mathbb{R}^{m}}$
+\end_inset
+
+ es compacto y, por ser además 
+\begin_inset Formula $d^{2}f(a)$
+\end_inset
+
+ definida positiva, existe 
+\begin_inset Formula $M>0$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\Phi(K)\geq M$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Como 
+\begin_inset Formula $\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}\rightarrow0$
+\end_inset
+
+ cuando 
+\begin_inset Formula $x\rightarrow a$
+\end_inset
+
+, existe 
+\begin_inset Formula ${\cal U}\in{\cal E}(a)$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in{\cal U},\left|\frac{2o(\Vert x-a\Vert)^{2}}{\Vert x-a\Vert^{2}}\right|<M$
+\end_inset
+
+, luego si 
+\begin_inset Formula $x\in{\cal U}\backslash\{a\},d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}>M-M=0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(x)>f(a)+\frac{1}{2}\Vert x-a\Vert^{2}\cdot0=f(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Obvio.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies4]$
+\end_inset
+
+ Fijamos 
+\begin_inset Formula $u\in\mathbb{R}^{m}$
+\end_inset
+
+ y definimos 
+\begin_inset Formula $\varphi(t):=a+tu$
+\end_inset
+
+ como la función 
+\begin_inset Formula $\varphi:\mathbb{R}\rightarrow\mathbb{R}^{m}$
+\end_inset
+
+ que parametriza la recta 
+\begin_inset Formula $a+<\vec{u}>$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula ${\cal U}\in{\cal E}(a)$
+\end_inset
+
+ con 
+\begin_inset Formula $f(a)\leq f(x)\forall x\in{\cal U}$
+\end_inset
+
+, si restringimos 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ a 
+\begin_inset Formula $\varphi^{-1}({\cal U})$
+\end_inset
+
+, un entorno de 0 en 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ alcanza un mínimo en 0, pues 
+\begin_inset Formula $(f\circ\varphi)(0)=f(\varphi(0))=f(a)\leq f(\varphi(t))\forall t\in\varphi^{-1}({\cal U})$
+\end_inset
+
+, y tenemos que 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ es de clase 
+\begin_inset Formula ${\cal C}^{2}$
+\end_inset
+
+ y semidefinida positiva.
+ Por la regla de la cadena, al ser 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ diferenciables,
+\begin_inset Formula 
+\begin{multline*}
+d(f\circ\varphi)(t)=df(\varphi(t))\circ d\varphi(t)\equiv\\
+\equiv\left(\begin{array}{ccc}
+\frac{\partial f}{\partial x_{1}}(a+tu) & \cdots & \frac{\partial f}{\partial x_{m}}(a+tu)\end{array}\right)\left(\begin{array}{c}
+u_{1}\\
+\vdots\\
+u_{m}
+\end{array}\right)=\sum_{i}\frac{\partial f}{\partial x_{i}}(a+tu)u_{i}
+\end{multline*}
+
+\end_inset
+
+Entonces 
+\begin_inset Formula $d^{2}(f\circ\varphi)(t)\equiv\frac{d}{dt}\sum_{i}\frac{\partial f}{\partial x_{i}}(a+tu)u_{i}=\sum_{i}u_{i}\frac{d}{dt}\left(\frac{\partial f}{\partial x_{i}}(a+tu)\right)$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $\frac{\partial f}{\partial x_{i}}$
+\end_inset
+
+ es diferenciable al ser 
+\begin_inset Formula $f$
+\end_inset
+
+ de clase 
+\begin_inset Formula ${\cal C}^{2}$
+\end_inset
+
+, 
+\begin_inset Formula 
+\begin{multline*}
+\frac{d}{dt}\left(\frac{\partial f}{\partial x_{i}}(a+tu)\right)=d\left(\frac{\partial f}{\partial x_{i}}\circ\varphi\right)(t)=d\frac{\partial f}{\partial x_{i}}(\varphi(t))\circ d\varphi(t)\equiv\\
+\equiv\left(\begin{array}{ccc}
+\frac{\partial^{2}f}{\partial_{i}\partial_{1}}(a+tu) & \cdots & \frac{\partial^{2}f}{\partial_{i}\partial_{m}}(a+tu)\end{array}\right)\left(\begin{array}{c}
+u_{1}\\
+\vdots\\
+u_{m}
+\end{array}\right)=\sum_{j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(a+tu)u_{j}
+\end{multline*}
+
+\end_inset
+
+Sustituyendo, 
+\begin_inset Formula $d^{2}(f\circ\varphi)(t)\equiv\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(a+tu)u_{i}u_{j}=d^{2}f(a+tu)(u,u)$
+\end_inset
+
+.
+ Pero al ser 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ una función real de una variable real dos veces derivable con su mínimo
+ en 0, sustituyendo 
+\begin_inset Formula $0\leq(f\circ\varphi)''(0)=d^{2}f(a)(u,u)$
+\end_inset
+
+, y como esto se cumple para todo 
+\begin_inset Formula $u\in\mathbb{R}^{m}$
+\end_inset
+
+, queda probado que 
+\begin_inset Formula $d^{2}f(a)$
+\end_inset
+
+ es semidefinida positiva.
+\end_layout
+
+\end_deeper
+\end_body
+\end_document
diff --git a/fvv1/n4.lyx b/fvv1/n4.lyx
new file mode 100644
index 0000000..07fa28a
--- /dev/null
+++ b/fvv1/n4.lyx
@@ -0,0 +1,592 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Podemos describir una región de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+De forma 
+\series bold
+implícita
+\series default
+, como el conjunto de puntos que cumplen 
+\begin_inset Formula $f(x)=0$
+\end_inset
+
+ para cierta función 
+\begin_inset Formula $f:{\cal U}\rightarrow\mathbb{R}^{k}$
+\end_inset
+
+, siendo 
+\begin_inset Formula ${\cal U}\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ un abierto.
+ La región 
+\begin_inset Formula $A=\{(x_{1},\dots,x_{n})\in{\cal U}:f(x_{1},\dots,x_{n})=0\}$
+\end_inset
+
+ está 
+\series bold
+descrita implícitamente de forma 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+-regular
+\series default
+ si 
+\begin_inset Formula $f$
+\end_inset
+
+ es de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall p\in A,\text{rg}(df(p))=k$
+\end_inset
+
+ (el rango de la diferencial es 
+\begin_inset Formula $k$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Enumerate
+De forma 
+\series bold
+paramétrica
+\series default
+, como la imagen de una función 
+\begin_inset Formula $\varphi:{\cal U}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+, siendo 
+\begin_inset Formula ${\cal U}\subseteq\mathbb{R}^{m}$
+\end_inset
+
+ un abierto.
+ La 
+\series bold
+parametrización
+\series default
+ es 
+\series bold
+
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+-regular
+\series default
+ si 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall p\in{\cal U},\text{rg}(d\varphi(p))=m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de la función implícita
+\series default
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Esto corresponde a FVV3, pero lo estudiamos por su utilidad práctica.
+\end_layout
+
+\end_inset
+
+ afirma que, para 
+\begin_inset Formula $A\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in A$
+\end_inset
+
+, existe un 
+\begin_inset Formula ${\cal U}\in{\cal E}(p)$
+\end_inset
+
+ tal que 
+\begin_inset Formula ${\cal U}\cap A$
+\end_inset
+
+ admite una presentación implícita 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+-regular si y sólo si existe 
+\begin_inset Formula ${\cal U}'\in{\cal E}(p)$
+\end_inset
+
+ tal que 
+\begin_inset Formula ${\cal U}\cap A$
+\end_inset
+
+ admite una parametrización 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+-regular.
+\end_layout
+
+\begin_layout Standard
+Sean pues 
+\begin_inset Formula ${\cal U}\subseteq\mathbb{R}^{m}\overset{\varphi}{\longrightarrow}{\cal V}\subseteq\mathbb{R}^{n}\overset{f}{\longrightarrow}{\cal W}\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ la parametrización y la forma implícita de 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $q\in{\cal U}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\varphi(q)=p\in{\cal V}$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $\text{Im}(d\varphi(q))=\ker(df(p))$
+\end_inset
+
+.
+ En efecto, como 
+\begin_inset Formula $f$
+\end_inset
+
+ es constante en 
+\begin_inset Formula $A$
+\end_inset
+
+ por ser 
+\begin_inset Formula $f(A)=\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ también lo es, luego 
+\begin_inset Formula $0=d(f\circ\varphi)(q)=df(p)\circ d\varphi(q)$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\text{Im}(d\varphi(q))\subseteq\ker(df(p))$
+\end_inset
+
+, pero como ambos subespacios tienen la misma dirección
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+(¿por qué?)
+\end_layout
+
+\end_inset
+
+, se tiene la igualdad.
+ Esto significa además que este espacio no depende de 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ o 
+\begin_inset Formula $f$
+\end_inset
+
+, y en esta situación llamamos 
+\series bold
+espacio tangente
+\series default
+ al compacto
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+¿por qué compacto?
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ en el punto 
+\begin_inset Formula $p$
+\end_inset
+
+ al espacio afín que pasa por 
+\begin_inset Formula $p$
+\end_inset
+
+ y tiene por dirección 
+\begin_inset Formula $\text{Im}(d\varphi(q))=\ker(df(p))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+gradiente
+\series default
+ en 
+\begin_inset Formula $a\in{\cal U}$
+\end_inset
+
+ de una función 
+\begin_inset Formula $f:D\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}$
+\end_inset
+
+ diferenciable en 
+\begin_inset Formula $a$
+\end_inset
+
+ al vector 
+\begin_inset Formula $\nabla f(a):=\left(\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}}\right)\in\mathbb{R}^{n}$
+\end_inset
+
+, la matriz de la diferencial de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ expresada como vector.
+ Para encontrar los extremos relativos de una función 
+\begin_inset Formula $f:D\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ sobre un subconjunto 
+\begin_inset Formula $D\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ no abierto:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ está dado en forma paramétrica como 
+\begin_inset Formula $\varphi({\cal U})$
+\end_inset
+
+, donde 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ es un abierto de 
+\begin_inset Formula $\mathbb{R}^{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi:{\cal {\cal U}}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es diferenciable, buscamos los extremos relativos de 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+, teniendo en cuenta que 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ tiene máximo absoluto en 
+\begin_inset Formula $a$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene un máximo absoluto en 
+\begin_inset Formula $\varphi(a)$
+\end_inset
+
+.
+ Si además 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es continua, un máximo relativo de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\varphi(a)$
+\end_inset
+
+ implica uno de 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+, y si 
+\begin_inset Formula $\varphi:({\cal U},{\cal T}_{u}|_{{\cal U}})\rightarrow(\varphi({\cal U}),{\cal T}_{u}|_{\varphi({\cal U})})$
+\end_inset
+
+ es abierta, un máximo relativo de 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula $a$
+\end_inset
+
+ es uno de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\varphi(a)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ está dado en forma implícita como 
+\begin_inset Formula $\{x\in{\cal U}:g(x)=0\}$
+\end_inset
+
+, donde 
+\begin_inset Formula ${\cal U}$
+\end_inset
+
+ es un abierto de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $g:{\cal U}\rightarrow\mathbb{R}^{k}$
+\end_inset
+
+ es de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+, aplicamos el 
+\series bold
+teorema de los multiplicadores de Lagrange
+\series default
+, que afirma que si 
+\begin_inset Formula $f:{\cal U}\rightarrow\mathbb{R}$
+\end_inset
+
+ es diferenciable, alcanza en un punto 
+\begin_inset Formula $a\in{\cal U}$
+\end_inset
+
+ un extremo relativo y 
+\begin_inset Formula $\text{rg}(dg(a))=k$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\nabla f(a)\in\text{span}(\nabla g_{1}(a),\dots,\nabla g_{k}(a)):=<\nabla g_{1}(a),\dots,\nabla g_{k}(a)>$
+\end_inset
+
+ (el espacio generado por los vectores).
+ 
+\series bold
+Demostración:
+\series default
+ Por el teorema de la función implícita, existen 
+\begin_inset Formula ${\cal V}\subseteq\mathbb{R}^{n-k}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula ${\cal W}\in{\cal E}(a)$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi:{\cal V}\rightarrow{\cal W}$
+\end_inset
+
+ de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{rg}(d\varphi(a))=n-k$
+\end_inset
+
+ tales que 
+\begin_inset Formula $D\cap{\cal W}=\varphi({\cal V})$
+\end_inset
+
+, y si 
+\begin_inset Formula $a$
+\end_inset
+
+ es extremo relativo de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+, por la continuidad de 
+\begin_inset Formula $\varphi$
+\end_inset
+
+, el punto 
+\begin_inset Formula $b\in{\cal V}$
+\end_inset
+
+ con 
+\begin_inset Formula $\varphi(b)=a$
+\end_inset
+
+ es extremo relativo de 
+\begin_inset Formula $f\circ\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula ${\cal V}$
+\end_inset
+
+, luego 
+\begin_inset Formula $d(f\circ\varphi)(b)=0=df(a)\circ\varphi(b)$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\text{Im}(d\varphi(b))=\ker(dg(a))=\ker(dg_{1}(a),\dots,dg_{k}(a))\subseteq\ker(df(a))$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\bigcap_{i=1}^{k}\ker(dg_{i}(a))\subseteq\ker(df(a))$
+\end_inset
+
+, que por un misterioso lema de álgebra equivale a que 
+\begin_inset Formula $df(a)\in\text{span}(dg_{1}(a),\dots,dg_{k}(a))$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+¿Ñandé?
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document