\part{Fundamentals}
\chapter{Topological linear spaces}


Functional analysis was originally developed to study \emph{functionals}; maps whose domain consisted of functions and whose codomain was $\mathbb{R}$ or $\mathbb{C}$. Eventually it was discovered that certain classes of linear spaces proved extremely effective at modelling function spaces (and therefore provided powerful tools to study functionals). Because of this, functional analysis eventually generalized to the study of \emph{complete topological linear spaces} and \emph{infinite dimensional linear spaces}; function spaces often tend to be linear spaces in both of these categories, so the specific study of functionals flows natrually from functional analysis.


Functional analysis regarding topological linear spaces is extremely compatible with real, complex, and vector analysis since the spaces of these disciplines ($\mathbb{R}^n$ and $\mathbb{C}^n$) are in fact finite dimensional topological linear spaces.


We will first look at \emph{topological linear spaces}, indeed, a topology on the linear space will be necessary in order to interpret series of vectors; without some topology there is no way to interpret convergence!


\begin{definition}[Topological linear space (TLS)]
A \emph{topological linear space (TLS)} is a linear space such that $F$ is a topological field, and there is a topology considered on $V$  such that $+,\cdot$ are continuous (taken as product topologies)
\end{definition}


We could indeed go down the path of studying the properties of TLSs with a generic topological space, however function spaces typically have a norm, 




Normed linear spaces form a TLS where the underlying topological space is a metric space; this makes normed linear spaces very desirable.

\begin{proposition}
Normed linear spaces induce a metric space.
\end{proposition}

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

\begin{proposition}
Normed linear spaces induce a translation and scaling invariant distance function.
Normed linear spaces are separable iff there is a countable subset whose span is dense in the space
\end{proposition}





\section{Banach spaces}

In mathematical analysis, $\mathbb{R}^n$ is particularly rich in mathematical analysis since it forms a complete space. For the sake of generalizing the properties of $\mathbb{R}^n$ and because completeness will ensure that we cannot use a vector series to converge objects that exist outside of the space. When studying infinite dimensional linear spaces, we rely on completeness so that the notion of the \emph{Schauder basis} works properly.

We have seen that normed linear spaces automatically introduce extremely nice topological properties (i.e forms a metric space), combining the notion of a norm and completeness results in a \emph{Banach space}; a complete normed linear space. Many function spaces are simply infinite dimensional Banach spaces, and indeed the familiar $\mathbb{R}^n$ is a Banach space.


\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}


Interesting, one can check that a normed linear space is Banach when absolute convergence implies convergence.
\begin{proposition}
Let $X$ be a normed linear space, X is a Banach space iff any absolute convergent series is a convergent series.
\end{proposition}





\section{Hilbert space}

\emph{Hilbert space}; complete inner product spaces.

\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}

Hilbert spaces are notorious as one of, if not, the closest class of abstract spaces to Euclidean space.