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Normed spaces

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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 \(V:=L_2(\bb{R}^d).\) (TODO: 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.