\chapter{Banach and Hilbert spaces}

We now focus on the deeper properties of such spaces.

\section{Hilbert spaces}




\begin{theorem}[Riesz representation theorem]
Let $H$ be a Hilbert space, then any continuous functional $T : H \to \mathbb{R}$ is of the form  $T\mathbf{x} = \langle \mathbf{x} , \mathbf{y} \rangle$
\end{theorem}

Since there is a bijective correspondence vectors and continuous functionals in Hilbert spaces (from a Hilbert space to its dual space), this gives rise to the Riesz map.
\[R : H \to H^{*}\]
\[R \mathbf{x} = T\]
\[T \mathbf{y} = \langle \mathbf{x} , \mathbf{y}\rangle\]



Within the finite dimension inner product spaces $\mathbb{C}^n$, it was seen that matrixes obey $\langle (\mathbf{A}^{\intercal})^{*} \mathbf{x} , \mathbf{y} \rangle = \langle \mathbf{x} , \mathbf{A} \mathbf{y}  \rangle$ (where $\mathbf{A}^{*}$ is the conjugate of $\mathbf{A}$). One may call $(\mathbf{A}^{\intercal})^{*}$ the \emph{adjoint of $\mathbf{A}$}; the matrix that arises when considering a matrix on the other argument, but keeping the inner product equal. 

Generalizing this notion from finite dimension inner product spaces to Hilbert spaces  gives the notion of an \emph{adjoint operator}.

\begin{definition}[Adjoint operator]
The \emph{adjoint operator of $T$} is the unique operator $T^{*}$ satisfying the following relation.
\[\langle T^{*}\mathbf{y} , \mathbf{x} \rangle_X = \langle \mathbf{y}, T \mathbf{x} \rangle_Y \]
\end{definition}

The fact that an adjoint operator of some $T$ is unique follows from the Riesz representation theorem.

As expected, the adjoint operator shares many properties with the Hermitian matrix.

\begin{itemize}
\item $(S+T)^{*} = S^{*} + T^{*}$
\item $T^{**}=T$
\item $I^{*}=I$
\item $(\lambda T)^{*}=\lambda^{*} T^{*}$
\item $(ST)^{*}= T^{*} S^{*}$
\item $\| T^{*}T \|= \| T \|^2$
\end{itemize}





In $\mathbb{C}^n$, the \emph{self-adjoint matrixes} are those that satisfy $\langle \mathbf{A} \mathbf{x} , \mathbf{y} \rangle = \langle \mathbf{x} , \mathbf{A} \mathbf{y}  \rangle$, or equivalently $\mathbf{A}= (\mathbf{A}^{\intercal})^{*}$. Similarly, we consider operators in Hilbert spaces that are their own adjoint operator.


\begin{definition}[Self-adjoint operator]
\[T^{*} = T\]
\end{definition}



\begin{proposition}
	\[ \mathrm{ker}(T^{*}) =  \mathrm{Im}(T)^{\perp} \]
\end{proposition}



\begin{proposition}
Let $T : X \to X$ be an operator invariant on $M$, then $T^{*}$ is invariant on $M^{\perp}$.
\end{proposition}

In the context of Hilbert spaces, one can discuss unitary isomorphisms, giving rise to unitary operators.


\begin{definition}[Unitary operator]
isomorphism whose adjoint operator is its inverse operator.
\[U^{-1}=U^{*}\]
\end{definition}

The Plancherel theorem states that the Fourier transform is unitary operator.

Here is a result that pertains not just to unitary operators, but any operator preserving the inner product.

\begin{proposition}
\[\langle \mathbf{x} ,  \mathbf{y} \rangle = \langle U\mathbf{x} , U \mathbf{y} \rangle  \implies \|U \mathbf{x} \| = \|\mathbf{x}\| \]
\end{proposition}




\subsection{Orthogonal projections}

In Hilbert space, both orthogonal projection and oblique projections exist; orthogonal projection is a Hermitian operator.

\begin{theorem}[Hilbert projection theorem]
Let $C$ be a closed convex subset of the Hilbert space $H$, then for any $\mathbf{x}$, there is some unique $\mathbf{x}_{*} \in C$ where the following holds for any $\mathbf{c} \in C$
\[\|\mathbf{x} - \mathbf{x}_{*}\| \leq \|\mathbf{x} - \mathbf{c}\| \]
Furthermore the map between $\mathbf{x}$ and its $\mathbf{x}_{*}$ is continuous
\end{theorem}


\begin{theorem}
Let $C$ be a closed convex subset of the Hilbert space $H$, then for any $\mathbf{x}$, there is some unique $\mathbf{x}_{*} \in C$ where the following holds for any $\mathbf{c} \in C$
\[ \mathbf{x} -\mathbf{x}_{*} \in C^{\perp} \iff \|\mathbf{x} - \mathbf{x}_{*}\| \leq \|\mathbf{x} - \mathbf{c}\| \]
\end{theorem}




\begin{theorem}
	\[  \|T\mathbf{x}_{*} - \mathbf{y}\| = \inf_{\mathbf{x} \in H_1} \|T\mathbf{x} - \mathbf{y}\| \iff  T^{*} T \mathbf{x}_{*} = T^{*} \mathbf{y}\]
\end{theorem}



\section{Banach spaces}


\subsection{Open mapping theorem}

An open mapping maps open sets to open sets.

\begin{theorem}[Open mapping theorem]
Let $T$ be a BLO between Banach spaces, then $T$ is an open mapping iff it is surjective
\end{theorem}


\subsection{Compact and Fredholm operators}



\subsection{Hahn-Banach theorem}

Existence theorem that guarantees an extension of a functional from a subspace to the entire space.

\begin{theorem}[Hahn-Banach theorem]
	Let $Y \leq X$, then every functional $f \in Y^{*}$ can be extended to some $g \in X^{*}$ such that $\|g\|_{X^{*}} =\|f\|_{Y^{*}}$
\end{theorem}

\subsection{Uniform boundedness theorem}

\begin{theorem}[Hahn-Banach theorem]
	Let $X$ be a Banach space and $Y$ be a normed linear space, and let $(T_i)_{i \in I}$ be a family of operators $T_i : X \to I$
	Then 
	\[\forall \mathbf{x} \in X [\exists C_{\mathbf{x}} \in \mathbb{R} [\forall i \in I [ \|T_i \mathbf{x}\| \leq C_{\mathbf{x}}]] ]  \implies \forall i \in I [ \| T_i \| \leq C]\]
\end{theorem}

\begin{corollary}
	Let $(T_n)_{n \in \mathbb{N}}$ such that $T_n \to T$ pointwise, then $T$ is linear and continuous, and $\|T\| \leq \mathrm{liminf}_{n \to \infty} \|T_n \|$
\end{corollary}


\chapter{Differentiation and integration}



\section{Frechet derivative}

Generalization of derivative to functions between normed linear spaces (over the same field).

when $\mathbf{h}$ in a neighborhood of $\mathbf{0}$
\[f(\mathbf{x} + \mathbf{h}) = f(\mathbf{x}) + f'(\mathbf{x})\mathbf{h} + o(\mathbf{h})\]

\section{The differential operator}




