Normed spaces

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 $T$ $$\begin{array}{rl}T:& V\to W\\ & v\mapsto Tv\end{array}$$

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 $\parallel \cdot \parallel :V\to \mathbb{R}$, such that, for $x,y\in V$ and $a\in \mathbb{F}$,

1. $\parallel x\parallel \ge 0$
2. $\parallel ax\parallel =|a|\parallel x\parallel$
3. $\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$

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