1-lecture. Normed space. Banach space lesson Plan

Theorem 1.3.2(Completeness). Every finite dimensional subspace of a normed space is complete. In particular, every finite dimensional normed space is complete.


Proof. We consider an arbitrary Cauchy sequence in and show that it is convergent in ; the limit will be denoted by . Let and any basis for . Then each has a unique representation of the form

Since is a Cauchy sequence, for every there is an such that

when . From this and Lemma 1.3.1 we have for some

where . Division by gives

This shows that each of the sequences

is Cauchy in or . Hence it converges; let denote the limit. Using these limits , we define

Clearly, . Furthermore,

On the right, . Hence , that is, . This shows that is convergent in . Since was an arbitrary Cauchy sequence in , this proves that is complete.∎

Theorem 1.3.3 (Closedness). Every finite dimensional subspace of a normed space is closed in .

Note that infinite dimensional subspaces need not be closed.

Example. Let and , where , so that is the set of all polynomials. is not closed in . (Why?) Another interesting property of a finite dimensional vector space is that all norms on lead to the same topology for , that is, the open subsets of are the same, regardless of the particular choice of a norm on . The details are as follows.