Normed spaces
2018-12-31 — 2019-01-04
Wherein the structure of vector spaces is recalled, norms are introduced to measure size and similarity, and completeness yielding Banach spaces as well as integral kernel operators on L2 are presented.
\[ \renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\mmm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}[1]{\mathbb{I}\{#1\}} \renewcommand{\inner}[2]{\langle #1,#2\rangle} \renewcommand{\Inner}[2]{\left\langle #1,#2\right\rangle} \renewcommand{\norm}[1]{\| #1\|} \renewcommand{\Norm}[1]{\|\langle #1\right\|} \renewcommand{\argmax}{\operatorname{arg max}} \renewcommand{\argmin}{\operatorname{arg min}} \renewcommand{\omp}{\mathop{\mathrm{OMP}}} \]
1 Vector space
A vector \(V\) space over \(\mathbb{F}\in\{\mathbb{C},\mathbb{R}\}\) is a set of objects that 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 arithmetic 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 \(V:=L_2(\mathbb{R}^d).\) (🚧TODO🚧 clarify \(L_2\) ) 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 that 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 a 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.