# Normed spaces

January 1, 2019 — January 4, 2019

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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 \(a\in\mathbb{F}\),

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

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