## Vector space

An vector space over is a set of objects which satisfy the rules of vector arithmetic, e.g. for all vectors , and all scalars , we have

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

### Linear integral operators

There is a type of operator that we use particularly often, defined by a *kernel*
and when is
an space of functions
(TODO: define here)
Specifically

## Normed space

An vector space over is a set of objects which satisfy the rules of vector arithmetic, e.g. for all vectors , and all scalars , we have

If is also an *normed space* over
then it is associated with a norm ,
such that, for and ,

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