# Normed spaces

January 1, 2019 — January 4, 2019

algebra
functional analysis
metrics
nonparametric

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## 1 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.

## 2 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}

### 2.1 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}

## 3 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 $$a\in\mathbb{F}$$,

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.