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 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.