updated probability and analysis

Mathematics/4th/Real_and_functional_analysis/Real_and_functional_analysis.tex

@@ -1172,7 +1172,10 @@
     \end{enumerate}
   \end{definition}
   \begin{definition}
-    Let $E$ be a normed vector space. The Banach space $E^*:=\mathcal{L}(E,\KK)$ is called \emph{dual space} of $E$.
+    Let $E$ be a normed vector space. The Banach space $E^*:=\mathcal{L}(E,\KK)$ is called \emph{dual space} of $E$. The \emph{bidual space} of $E$ is $E^{**}:=(E^*)^*$.
+  \end{definition}
+  \begin{definition}
+    Let $E$ be a normed vector space. We say that $E$ is \emph{reflexive} if $E=E^{**}$.
   \end{definition}
   \subsubsection{Compact operators}
   \begin{definition}
@@ -1265,7 +1268,7 @@
     \end{enumerate}
   \end{definition}
   \begin{theorem}[Hahn-Banach theorem]
-    Let $E$ be a real vector space, $F\subseteq E$ be a subspace, $p:E\rightarrow\RR$ be a convex functional and $u\in F^*$. If $u(z)\leq p(z)$ $\forall z\in F$, then $\exists v\in E^*$ such that $v(z)=u(z)$ $\forall z\in F$ and $v(x)\leq p(x)$ $\forall x\in E$ ($v$ is called a \emph{prolongation} of $u$).
+    Let $E$ be a real vector space, $F\subseteq E$ be a subspace, $p:E\rightarrow\RR$ be a convex functional and $u\in F^*$. If $u(z)\leq p(z)$ $\forall z\in F$, then $\exists v\in E^*$ such that $v(z)=u(z)$ $\forall z\in F$ and $v(x)\leq p(x)$ $\forall x\in E$ ($v$ is called an \emph{extension} of $u$).
   \end{theorem}
   \begin{definition}[Seminorm]
     Let $E$ be a vector space over $\KK$. A \emph{seminorm} $p:E\rightarrow[0,\infty)$ is a functional that satisfies:
@@ -1278,12 +1281,12 @@
     Let $E$ be a vector space over $\KK$. A norm defined on $E$ is a seminorm.
   \end{lemma}
   \begin{theorem}[Hahn-Banach theorem]
-    Let $E$ be a vector space over $\KK=\RR,\CC$, $F\subseteq E$ be a subspace, $p:E\rightarrow\RR$ be a seminorm and $u\in F^*$. If $\abs{u(z)}\leq p(z)$ $\forall z\in F$, then $\exists v\in E^*$ such that $v(z)=u(z)$ $\forall z\in F$ and $\abs{v(x)}\leq p(x)$ $\forall x\in E$. That is, $v$ prolongates $u$.
+    Let $E$ be a vector space over $\KK=\RR,\CC$, $F\subseteq E$ be a subspace, $p:E\rightarrow\RR$ be a seminorm and $u\in F^*$. If $\abs{u(z)}\leq p(z)$ $\forall z\in F$, then $\exists v\in E^*$ such that $v(z)=u(z)$ $\forall z\in F$ and $\abs{v(x)}\leq p(x)$ $\forall x\in E$. That is, $v$ extends $u$.
   \end{theorem}
   \begin{theorem}[Hahn-Banach theorem]
     Let $E\ne\{0\}$ be a normed vector space.
     \begin{enumerate}
-      \item If $F\subseteq E$ is a subspace and $u\in F^*$, then $\exists v\in E^*$ such that prolongates $u$ and $\norm{v}=\norm{u}$.
+      \item If $F\subseteq E$ is a subspace and $u\in F^*$, then $\exists v\in E^*$ such that extends $u$ and $\norm{v}=\norm{u}$.
       \item For all $a\in E$, $\exists v\in E^*$ such that $v(a)=\norm{a}$ and $\norm{v}=1$.
       \item If $F\subseteq E$ is a closed subspace and $a\in E\setminus F$, then $\exists v\in E^*$ such that $v(a)=1$ and $v(F)=\{0\}$.
     \end{enumerate}
@@ -1306,6 +1309,15 @@
   \begin{proposition}
     Let $E$, $F$ be normed vector spaces. The function $$\function{}{\mathcal{L}(E,F)}{\mathcal{L}(F^*,E^*)}{T}{T^*}$$ is linear, bijective and isometric. That is, $\norm{T}=\norm{T^*}$.
   \end{proposition}
+  \begin{theorem}
+    Let $\Omega\subseteq \RR^n$ be a measurable set, $1\leq p\leq \infty$ and $q$ be the Hölder conjugate of $p$. Then:
+    \begin{itemize}
+      \item If $1<p<\infty$, then ${(L^p(\Omega))}^*=L^q(\Omega)$.
+      \item If $p=1$, then ${(L^1(\Omega))}^*=L^\infty(\Omega)$.
+      \item If $p=\infty$, then ${(L^\infty(\Omega))}^*\supsetneq L^1(\Omega)$.
+    \end{itemize}
+    In particular, for $1<p<\infty$, $L^p(\Omega)$ is reflexive, while $L^1(\Omega)$ and $L^\infty(\Omega)$ are not.
+  \end{theorem}
   \subsubsection{Spectrum and eigenvalues}
   \begin{proposition}
     Let $E$ be a Banach space and $T\in\mathcal{L}(E)$. Then, $\forall \alpha\in\KK$, $\im(T-\alpha\id)$ and $\ker(T-\alpha\id)$ are invariant over $T$. Moreover, if $\alpha\ne 0$, the function $$\function{S}{\ker(T-\alpha\id)}{\ker(T-\alpha\id)}{x}{\alpha x}$$ is an isomorphism.
@@ -1372,7 +1384,7 @@
       \item The limit $Tx:=\displaystyle\lim_{n\to\infty}T_nx$ exists $\forall x\in D\subseteq E$, where $D$ is a dense set in $E$.
       \item The sequence $(T_nx)$ is bounded $\forall x\in E$.
     \end{itemize}
-    Then, $T$ can be prolongated into a bounded operator such that: $$\norm{T}\leq \liminf_{n\to\infty}\norm{T_n}$$
+    Then, $T$ can be extended into a bounded operator such that: $$\norm{T}\leq \liminf_{n\to\infty}\norm{T_n}$$
   \end{corollary}
   \subsection{Hilbert spaces}
   \subsubsection{Inner products}

Mathematics/5th/Advanced_probability/Advanced_probability.tex

@@ -221,5 +221,57 @@
       \Exp(f(X)\mid Y)=\int_\RR{f(x)\dd{\mathcal{L}^{X\mid Y}(Y,x)}}
     $$
   \end{theorem}
+  \subsection{Martingales}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob)$ be a probability space. A \emph{filtration} is a sequence of sub-$\sigma$-algebras ${(F_n)}_{n\in\NN}$ such that $\forall n\in\NN$, $F_n\subseteq F_{n+1}$. The tuple $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ is called a \emph{filtered probability space}.
+  \end{definition}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space. A stochastic process ${(X_n)}_{n\in\NN}$ is \emph{adapted} to ${(F_n)}_{n\in\NN}$ if $\forall n\in\NN$, $X_n$ is $F_n$-measurable.
+  \end{definition}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space and ${(M_n)}_{n\in\NN}$ be an adapted stochastic process. We say that ${(M_n)}_{n\in\NN}$ is a \emph{martingale} if $\forall n\in\NN$, $\Exp(\abs{M_n})<\infty$ and $\Exp(M_{n+1}\mid F_n)=M_n$.
+  \end{definition}
+  \begin{remark}
+    A \emph{submartingale} and \emph{supermartingale} are defined similarly, but with $\Exp(M_{n+1}\mid F_n)\geq M_n$ and $\Exp(M_{n+1}\mid F_n)\leq M_n$ respectively.
+  \end{remark}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space. A \emph{stopping time} is a random variable $\tau:\Omega\to \NN\cup\{\infty\}$ such that $\forall n\in\NN$, $\{\tau\leq n\}\in F_n$.
+  \end{definition}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space, $\tau$ be a stopping time and $M:={(M_n)}_{n\in\NN}$ be a process. We define the \emph{stopped process} $M^\tau:={(M_{\tau\wedge n})}_{n\in\NN}$.
+  \end{definition}
+  \begin{proposition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space, $\tau$ be a stopping time and $M:={(M_n)}_{n\in\NN}$ be a martingale. Then, $M^\tau:={(M_{\tau\wedge n})}_{n\in\NN}$ is a martingale.
+  \end{proposition}
+  \begin{corollary}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space, $\tau$ be a bounded stopping time and $M:={(M_n)}_{n\in\NN}$ be a martingale. Then, $\Exp(M_\tau)=\Exp(M_0)$.
+  \end{corollary}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space, $\tau$ be a stopping time. We define:
+    $$
+      \mathcal{F}_\tau:=\{A\in\mathcal{F}:\forall n\in\NN, A\cap \{\tau=n\}\in F_n\}
+    $$
+  \end{definition}
+  \begin{remark}
+    It can be seen that in the above definition, $\mathcal{F}_\tau$ is a $\sigma$-algebra.
+  \end{remark}
+  \begin{proposition}
+    Let $(\Omega,\mathcal{F},\Prob,{(F_n)}_{n\in\NN})$ be a filtered probability space, $\rho\leq\tau$ be two bounded stopping times, and $M$ be a martingale. Then, $\Exp(M_\tau\mid \mathcal{F}_\rho)=M_\rho$.
+  \end{proposition}
+  \begin{theorem}
+    Let $(\Omega,\mathcal{F},\Prob)$ be a probability space and ${(M_n)}_{n\in\NN}$ be a martingale such that $\sup_{n\in\NN}{\Exp(\abs{M_n})}<\infty$. Then, there exists a random variable $M_\infty$ such that $M_n\overset{\text{a.s.}}{\to} M_\infty$.
+  \end{theorem}
+  \begin{definition}
+    Let $(\Omega,\mathcal{F},\Prob)$ be a probability space. A family of random variables ${(X_t)}_{t\in I}$ is said to be \emph{uniformly integrable} if:
+    $$
+      \lim_{a\to\infty}{\sup_{t\in I}{\Exp(\abs{X_t}\indi{\abs{X_t}>a})}}=0
+    $$
+  \end{definition}
+  \begin{proposition}
+    Let $(\Omega,\mathcal{F},\Prob)$ be a probability space and ${(X_n)}_{n\in\NN}$ be a family of uniformly integrable random variables such that $X_n\overset{\text{a.s.}}{\to} X$. Then, $X$ is integrable and $\Exp(X_n)\to \Exp(X)$.
+  \end{proposition}
+  \begin{theorem}
+    Let $(\Omega,\mathcal{F},\Prob)$ be a probability space and ${(M_n)}_{n\in\NN}$ be a martingale bounded in $L^p$, $1<p<\infty$. Then, there exists a random variable $M_\infty$ such that $M_n\overset{L^p}{\to} M_\infty$ and $M_n\overset{\text{a.s.}}{\to} M_\infty$.
+  \end{theorem}
 \end{multicols}
 \end{document}

Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex

@@ -3,6 +3,57 @@
 \begin{document}
 \changecolor{ATFAPDE}
 \begin{multicols}{2}[\section{Advanced topics in functional analysis and PDEs}]
+  \subsection{\texorpdfstring{$L^p$}{Lp} spaces}
+  \subsubsection{Topologies of \texorpdfstring{$L^p$}{Lp} spaces}
+  \begin{definition}
+    Let $E$ be a Banach space and ${(x_n)}_{n\in\NN}\in E$. We say that ${(x_n)}_{n\in\NN}$ \emph{converges weakly} to $x\in E$ if $\forall f\in E^*$ we have:
+    $$
+      \lim_{n\to\infty}{f(x_n)}=f(x)
+    $$
+    We denote this by $x_n\rightharpoonup x$.
+  \end{definition}
+  \begin{definition}
+    Let $E$ be a Banach space and ${(x_n)}_{n\in\NN}\in E$. We say that ${(x_n)}_{n\in\NN}$ \emph{converges strongly} to $x\in E$ if $\forall f\in E^*$ we have:
+    $$
+      \lim_{n\to\infty}{\norm{x_n-x}}=0
+    $$
+    We denote this by $x_n\to x$.
+  \end{definition}
+  \begin{definition}
+    Let $E$ be a Banach space and ${(x_n)}_{n\in\NN}\in E$. Assume $E=F^*$, where $F$ is a Banach space. Then, we say that ${(x_n)}_{n\in\NN}$ \emph{converges weakly-*} to $x\in E$ if $\forall f\in F$ we have:
+    $$
+      \lim_{n\to\infty}{x_n(f)}=x(f)
+    $$
+    We denote this by $x_n\overset{*}\rightharpoonup x$.
+  \end{definition}
+  \begin{theorem}
+    Let $E$ be a Banach space and ${(x_n)}_{n\in\NN}\in E$. Then:
+    \begin{enumerate}
+      \item If $x_n\to x$, then $x_n\rightharpoonup x$.
+      \item If $E$ is reflexive then weak convergence is equivalent to weak-* convergence.
+      \item If $x_n\to x$, $x_n\rightharpoonup x$ or $x_n\overset{*}\rightharpoonup x$, then $(x_n)$ is bounded.
+      \item Let ${(L_n)}_{n\in\NN}\in E^*$. If $x_n\to x$ in $E$ and $L_n\overset{*}\rightharpoonup L$ in $E^*$, then $L_n(x_n)\to L(x)$ in $\CC$.
+    \end{enumerate}
+  \end{theorem}
+  \begin{theorem}[Banach-Alaoglu theorem]
+    Let $\Omega\subseteq \RR^d$ be a set and $1<p<\infty$. If $(f_n)$ is a bounded sequence in $L^p(\Omega)$, then there is a subsequence $(f_{n_k})$ and $f\in L^p(\Omega)$ so that $f_{n_k}\rightharpoonup f$ in $L^p(\Omega)$. If $p=\infty$, then there is a subsequence $(f_{n_k})$ and $f\in L^\infty(\Omega)$ so that $f_{n_k}\overset{*}\rightharpoonup f$ in $L^\infty(\Omega)$.
+  \end{theorem}
+  \subsubsection{Lower-semicontinuous functions and convexity}
+  \begin{definition}
+    Let $E$ be a Banach space, $(x_n)\in E$ and $f:E\to\RR$. We say that $f$ is \emph{strongly lower-semicontinuous} if:
+    $$
+      x_n\to x\implies f(x)\leq \liminf_{n\to\infty}{f(x_n)}
+    $$
+  \end{definition}
+  \begin{remark}
+    Analogously, we can define \emph{weakly lower-semicontinuity} and \emph{weak-* lower-semicontinuity} by replacing $x_n\to x$ by $x_n\rightharpoonup x$ and $x_n\overset{*}\rightharpoonup x$ respectively.
+  \end{remark}
+  \begin{theorem}
+    Let $E$ be a Banach space and $f:E\to\RR$ be convex. Then, $f$ is strongly lower-semicontinuous if and only if $f$ is weakly lower-semicontinuous. In particular, the map ${\norm{\cdot}}_E$ is weakly lower-semicontinuous.
+  \end{theorem}
+  \begin{theorem}
+    Let $\Omega\subseteq \RR^d$ be a set and $1<p<\infty$. Then if $f_n\rightharpoonup f$ in $L^p(\Omega)$, then: $$\norm{f}_p\leq \liminf_{n\to\infty}{\norm{f_n}_p}$$ In addition, if $\displaystyle\norm{f}_p=\lim_{n\to\infty}{\norm{f_n}_p}$, then $f_n\to f$ in $L^p(\Omega)$.
+  \end{theorem}
   \subsection{Sobolev spaces}
   \begin{definition}[Sobolev spaces]
     Let $\Omega\subseteq \RR^d$ be an open set, $m\in\NN$ and $1\leq p\leq \infty$. We define the \emph{Sobolev spaces} $W^{m,p}$ as:

README.md

@@ -1,6 +1,6 @@
 # Mathematics and Physics summaries
 
-Summary of each subject in Mathematics and Physics degree at UAB (Universitat Autònoma de Barcelona).
+Summary of each subject in Mathematics and Physics degree at UAB (Universitat Autònoma de Barcelona) at Paris Dauphine - PSL University.
 
 ### TODO
 

index.html

@@ -13,7 +13,7 @@
   <div class="header">
     <h1 class="special-color">Mathematics and Physics notes</h1>
     <br>
-    <p>This website intends to summarize some notes about each <q>important</q> subject in Mathematics and Physics degree at Autonomous University of Barcelona (UAB).</p>
+    <p>This website intends to summarize some notes about each <q>important</q> subject in Mathematics and Physics degree at Autonomous University of Barcelona (UAB) and at Paris Dauphine - PSL University.</p>
     <p>You can download the sections separately as you like or the whole file of Mathematics or Physics (better in order to avoid linking errors).</p>
   </div>
   <div class="row">