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

## Vector space

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

Just doing vector arithemetic is not usually intersting 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 \(V:=L_2(\bb{R}^d).\) (TODO: define \(L_2\) here) Specifically \[\begin{aligned}T:&V\to V\\
(Tv)(x)&\mapsto \int_{\bb{R}^d}K(x,y)v(y)\dd y
\end{aligned}\]

## Normed space

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

If \(V\) is also an *normed space* over \(\bb{F}\) 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*.