1-lecture. Normed space. Banach space lesson Plan

Lemma (Translation invariance). A metric induced by a norm on a normed space satisfies

(1.1.6),

for all and every scalar .

Proof. We have

and



1.2. FURTHER PROPERTIES OF NORMED SPACES
By definition, a subspace of a normed space is a subspace of considered as a vector space, with the norm obtained by restricting the norm on to the subset . This norm on is said to be induced by the norm on . If is closed in , then is called a closed subspace of .

By definition, a subspace of a Banach space is a subspace of considered as a normed space. Hence we do not require to be complete.

Theorem 1.2.1. (Subspace of a Banach space). A subspace Y of a Banach space is complete if and only if the set is closed in .


Convergence of sequences and related concepts in normed spaces follow readily from the corresponding definitions for metric spaces and the fact that now
:

(i) A sequence in a normed space is convergent if contains an such that

Then we write and call the limit of .

(ii) A sequence in a normed space is Cauchy if for every there is an such that

(1.2.1)

Sequences were available to us even in a general metric space. In a normed space we may go an important step further and use series as follows.

Infinite series can now be defined in a way similar to that in calculus. In fact, if is a sequence in a normed space , we can associate with the sequence of partial sums

where . If is convergent, say,

then the infinite series or, briefly, series

(1.2.2)

is said to converge or to be convergent, is called the sum of the series and we write

If converges, the series (1.2.2) is said to be absolutely convergent. However, we warn the reader that in a normed space , absolute convergence implies convergence if and only if is complete.

The concept of convergence of a series can be used to define a "basis" as follows. If a normed space contains a sequence with the property that for every there is a unique sequence of scalars such that

then is called a Schauder basis (or basis) for . The series

which has the sum is then called the expansion of with respect to , and we write