\part{Fundamentals}


Readers will be familiar with mathematical analysis; the branch of mathematics that studies functions. Different spaces have their own unique set of properties, and so therefore functions over different spaces often display different behaviours. For complex functions, holomorphic functions are analytic, while for real functions differentiable in some neighborhood are not necessarily analyitic.

Originally the term 'functional' was used for a map whose domain consisted of functions and whose codomain was $\mathbb{R}$ or $\mathbb{C}$. The term has a more liberal definition now, relaxing the domain of a functional to be a linear space rather than specifically a function space.

Functional is therefore just a synonym for linear forms.

Since $\mathbb{R}^n$ are linear spaces, functional analysis can be used as a generalization of real and vector analysis, 

Though perhaps the most interesting use of functional analysis is to study function spaces; many function spaces of interest are indeed linear spaces. This allows us to do things like study differentiation and integration as functionals, or consider the Fourier series to be some closed linear span! This was the initial motivation for functional analysis.

\chapter{Banach and Hilbert spaces}

Linear spaces are only an algebraic structure, so in order to study ideas of mathematical analysis on a linear space we require these spaces to also have some topological structure.

%Definition; Topological linear space (TLS)

We are already familiar with some TLSs; the \emph{normed linear spaces and inner product spaces}. When studying analysis on $\mathbb{R}^n$, the norm is used in the definition for limits in these spaces; the norm produces a metric space that allows such things to exist.

Linear spaces equipped with a norm are special because not only do they form a TLS, but the underlying topological spaces is a metric space; giving this TLS many rich properties.
%Proposition; Normed linear spaces induce a metric space.

Furthermore, the properties of linear spaces give the underlying metric space another nice property.

%Proposition; Normed linear spaces induce a translation and scaling invariant distance function.

%Discuss the desire for this induced metric space to be complete so that limits work the same as real/vector analysis. hint to banach spaces

\section{Banach spaces}
\begin{definition}[Banach space]
A \emph{Banach space} is a normed linear space $(V,\| \cdot \|)$ such that its induced metric space $(V,d)$ (where $d(\mathbf{u},\mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|$) is a complete metric space.
\end{definition}


%Proposition; In a Banach space, absolute convergent series are convergent series.
%Discuss the failure of using linear independence, linear spans, and Hamel basis' for infinite dimensional TLSs


%Definition; Infinite linear independence
%Definition; Closed linear span
%Definition; Schauder basis




\section{Hilbert space}
% What about a TLS with an inner product?

\begin{definition}[Hilbert space]
A \emph{Hilbert space} is an inner product space $(V,\langle \cdot  , \cdot \rangle)$ such that its induced metric space $(V,d)$ (where $d(\mathbf{u},\mathbf{v}) = \sqrt{\langle \mathbf{u} - \mathbf{v} , \mathbf{u} - \mathbf{v} \rangle}$) is a complete metric space.
\end{definition}

\chapter{$L^p$ spaces}
L^p space (Lebesgue integrable functions with finite p-norm)
p-norm
Young's inequality
Hölder's inequality
Minkowski's inequality
\section{Othogonal Schauder basis of an L^p spaces}
%Legendre polynomials
%Chebyshev polynomials
%Hermite polynomials
%Orthogonality of trigonometric functions
\begin{theorem}[Riesz-Fischer theorem]
\[L^2 \text{ is a Hilbert space}\]
\end{theorem}
%L^2 spaces are Hilbert spaces

\chapter{C^k spaces}
C^0 space (continuous functions)
C^k space (k times continuously differentiable functions)
C^\infty space (infinitely differentiable functions)
C^\omega space (analytic functions)

\chapter{Operator theory}
\begin{definition}[Operator]
An \emph{operator} is a continuous linear map between linear spaces with the same field $T : X \to Y$.
An \emph{operator} is a map between linear spaces $T : X \to Y$. The term is often used when the domain is a function space.
\end{definition}


\begin{definition}[Functional]
A \emph{functional} is a continuous linear map between normed linear space and its field $T : X \to F$.
A \emph{functional} is a map between a linear space and its field $T : X \to F$.
\end{definition}

\begin{definition}[Projection]
A \emph{projection} is an idempotent operator.
\[P^2=P\]
\end{definition}

An operation is idempotent if subsequent compositions of the operation do not change its result.


\begin{proposition}
Continuous linear operators between normed linear spaces are Lipschitz continuous.
\end{proposition}

This motivates the following definition.
%Definition; Bounded linear operator

\begin{proposition}
Let $T$ be a bijective bounded linear operator, then $T^{-1}$ is linear.
Let $T$ be a bijective bounded linear operator, then $T^{-1}$ is continuous if $\|\mathbf{x}\| \leq c \|T\mathbf{x}\|$.
\end{proposition}


Since all operators are Lipschitz in normed linear spaces, every operator has some smallest Lipschitz constant. We call this constant the \emph{operator norm}.

\begin{definition}[Operator norm]
\[\|T\| = \sup_{\mathbf{x} \neq \mathbf{0}} \frac{\|T\mathbf{x}\|}{\mathbf{x}}\]
\end{definition}

As it turns out (and I not-so-suavely alluded to by calling it the operator NORM) operator norms are indeed a norm! We denote these operator spaces by B(X,Y)

When the codomain of the operator is a complete space, the operator space is also complete.

Compositions of operators are operators
\[\|ST\| \leq \|S\|\|T\|\]

\section{Matrix norms}
Frobenius norm


- differentiation operator
- differential operator
- adjoint operator
- bounded linear operator
- operator norm


- Hahn-Banach theorem
- integral linear transform
### examples of integral linear operators
- Laplace transform
- Fourier transform
- Hankel transform
- Mellin transform
\chapter{Banach algebrae}
Banach algebra; Associative algebra on a Banach space
C-Algebra
C\*-Algebra
Von Neumann algebra

\part{Advanced}
\chapter{Spectral theory}
\chapter{More function spaces?}
H^p space (Holomorphic functions with finite p-norm)
\section{Fréchet space (functional analysis)}


















<!DOCTYPE html>
<html>
	<head>
		<title>Functional analysis</title>
		<?php echo file_get_contents("/srv/http/head.php") ?>

			<h1>Functional analysis</h1>


			<hr>

			<ul>
				<li><a href='/daigaku/2022_aut/dm.php'>Discrete Mathematics</a></li>
				<li><a href='/daigaku/2023_spr/ra.php'>Real Analysis</a></li>
				<li><a href='/daigaku/2024_spr/lifa.php'>Lebesgue Integration and Fourier Analysis</a></li>
			</ul>







			<h3>Linear space Spazio lineare</h3>
			<p>4-tuple \( (V,F,+,\cdot ) \) such that:</p>
			<ul>
				<li>\(V\) is the set of linear objects</li>
				<li>\(F\) is the scalar field (often made as \(\mathbb{R} \) or \(\mathbb{C}\))</li>
				<li>\(+\) is the linear addition operator defined on \(V\)</li>
				<li>\(\cdot\) is the scalar multiplication operator defined between \(V\) and \(F\)</li>
			</ul>
			<p>\((V,F,+, \cdot) \text{ is a linear space } \iff \textbf{v},\textbf{u} \in V \land c,d \in F \implies\)</p>
			<ul>
				<li>\(\textbf{v}+\textbf{u}=\textbf{u}+\textbf{v}\)</li>
				<li>\( (\textbf{v}+\textbf{u})+\textbf{w} = \textbf{v}+(\textbf{u}+\textbf{w}) \) </li>
				<li>\( c(d\textbf{v}) = (cd)\textbf{v} \) </li>
				<li>\( c(\textbf{v}+\textbf{u}) = c\textbf{v} + c\textbf{u} \)</li>
				<li>\( (c+d)\textbf{v} = c\textbf{v} + d\textbf{u} \) </li>
				<li>\( \exists 1 : 1 \cdot \textbf{v} = \textbf{v} \) </li>
				<li>\( \exists \textbf{0} : \textbf{v} + \textbf{0} = \textbf{v} \)</li>
				<li>\( \exists (-\textbf{v}) : \textbf{v}+(-\textbf{v})=\textbf{0} \) </li>
			</ul>
			<h4>Normed linear space</h4>
			<p>\( (V , \| \cdot \| ) \)</p>
			<h4>Inner product space</h4>
			<p>\( (V , \langle \cdot \rangle ) \)</p>

			<h3>\(L^{p}\) space</h3>
			<p>Function space defined by the following:</p>
			<p>\(L^p = \{ f : (\int |f|^p d \lambda)^{\frac{1}{p}} \lt \infty \}\)</p>
			<p>\(L^{\infty} = \{ f : f \text{ is bounded a.e.} \}\)</p>
			<p>These spaces have \( \text{dim}(X) = \infty\) since a homomorphism of functions defined on \(\mathbb{R}\) can be made to vectors of uncountable dimension. See Linear Algebra</p>
			<h4>\(p\)-norm</h4>
			<p>Quasi-norm (function similar to a norm but not exactly) defined on \(L^p\) spaces as such:</p>
			<p>\(\| f \|_{p} = (\int |f|^p d \lambda )^{\frac{1}{p}}\)</p>
			<p>\(\| f \|_{\infty} = \sup |f(x)| \)</p>

			<h3>Euclidean space</h3>
			<p>Inner product linear space on a finite dimensional set of rational, real, or complex numbers</p>
			<p>\(E = (V, \cdot )\)</p>


			<h3>Hilbert spaces</h3>
			<p>Complete, inner product linear space. These spaces always have an orthonormal basis</p>
		All inner product spaces have an orthonormal basis.

			<h4>Separability</h4>
			<p>\( (H,\langle \cdot \rangle ) \text{ is separable } \iff \text{ orthonormal basis is countable}\)</p>

















\begin{definition}[Chebyshev polynomials]
The \emph{Chebyshev polynomials (first kind)} are a sequence of polynomials $(T_n)_{n \in \mathbb{N}}$ relating to cosine relations of an angle with factor $n$.
\[ T_n ( \cos(\theta) ) = \cos (n \theta) \]
The \emph{Chebyshev polynomials (second kind)} are a sequence of polynomials $(U_n)_{n \in \mathbb{N}}$ relating to sine relations of an angle with factor $n$.
\[  U_{n-1} ( \cos(\theta) ) = \frac{\sin (n \theta)}{\sin (\theta)} \]
\end{definition}



			<ul>
				<li>\( T_{0} = 1 \)</li>
				<li>\( T_{1} = x \)</li>
				<li>\( T_{n+1} = 2x T_{n} - T_{n-1} \)</li>
			</ul>
			<ul>
				<li>\( U_{0} = 1 \)</li>
				<li>\( U_{1} = 2x \)</li>
				<li>\( U_{n+1} = 2x U_{n} - U_{n-1} \)</li>
			</ul>

\begin{definition}[Legendre polynomials]
The \emph{Legendre polynomials} are a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ forming an orthogonal basis for $L^2 [-1,1]$ with weight 1.
\end{definition}
