\(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}[1]{\mathbb{I}\{#1\}} \renewcommand{\inner}[2]{\langle #1,#2\rangle} \renewcommand{\Inner}[2]{\left\langle #1,#2\right\rangle} \renewcommand{\norm}[1]{\| #1\|} \renewcommand{\Norm}[1]{\|\langle #1\right\|} \renewcommand{\argmax}{\operatorname{arg max}} \renewcommand{\argmin}{\operatorname{arg min}} \renewcommand{\omp}{\mathop{\mathrm{OMP}}}\)

## Vector space

A vector \(V\) space over \(\mathbb{F}\in\{\mathbb{C},\mathbb{R}\}\) is a set of objects which satisfy the rules of vector arithmetic, e.g. for all vectors \(x,y\in V\), and all scalars \(\alpha,\beta\in\mathbb{F}\), we have \(\alpha x + \beta y\in V.\)

Just doing vector arithemetic is not usually interesting for our purposes; we usually wish to measure what our vectors are doing in some sense, and so need a notion of similarity etc. This is where norms come in.

## Operators

a.k.a. mappings, translations. These are simply functions from one vector space to another. We can write the operator \(T\) \[\begin{aligned} T:&V\to W\\ &v \mapsto Tv \end{aligned}\]

### Linear integral operators

There is a type of operator that we use particularly often, defined by a *kernel*
\[K:V \times V \to \mathbb{C}\]
and when \(V\) is
an \(L_2\) space of functions
\(V:=L_2(\mathbb{R}^d).\)
(🏗 define \(L_2\) )
Specifically
\[\begin{aligned}T:&V\to V\\
(Tv)(x)&\mapsto \int_{\mathbb{R}^d}K(x,y)v(y)\dd y
\end{aligned}\]

## Normed space

A vector \(V\) space over \(\mathbb{F}\in\{\mathbb{C},\mathbb{R}\}\) is a set of objects which satisfy the rules of vector arithmetic, e.g. for all vectors \(x,y\in V\), and all scalars \(\alpha,\beta\in\mathbb{F}\), we have \(\alpha x + \beta y\in V.\)

If \(V\) is also an *normed space* over \(\mathbb{F}\)
then it is associated with a norm \(\|\cdot\|:V\to \mathbb{R}\),
such that, for \(x,y\in V\) and \(a\in\mathbb{F}\),

- \(\norm{x}\geq 0\)
- \(\norm{ax}=|a|\norm{x}\)
- \(\norm{x+y}\leq \norm{x}+\norm{y}\)

If \(V\) is complete (i.e. closed under limits)
then it is called a *Banach space*.

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