# Normed spaces

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

## 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 $$V:=L_2(\bb{R}^d).$$ (🏗 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 $$a\in\mathbb{F}$$,

1. $$\norm{x}\geq 0$$
2. $$\norm{ax}=|a|\norm{x}$$
3. $$\norm{x+y}\leq \norm{x}+\norm{y}$$

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