The Living Thing / Notebooks :

Normed spaces

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.