Banach Spaces

Banach Space

A normed space is a Banach space if it is complete as a metric space.

e.g.

• Let be a domain and . A function is said to belong to the Hölder space if
  • The first term measures the boundedness of .
  • The second term measures the -Hölder continuity of .
    Equipped with this norm, is a Banach space.
• Suppose is a measure space, then is a Banach space with norm . The proof of completeness will be provided later here after we have established a tool for proving completeness of normed spaces.
  

Absolutely Convergent Series

A series in a normed space is said to be absolutely convergent if the series converges in .

Theorem

A normed space is Banach if and only if every absolutely convergent series in it converges.

Proof Suppose is a Banach space, and is absolutely convergent, then converges in . So the sequence of partial sum of norms is Cauchy in . Consider , , for any , we have so is Cauchy in , and hence convergent since is complete.
Conversely, we pick a Cauchy sequence in , say . We can pick a subsequence such that . Define , and for all . Then the series is absolutely convergent because and hence convergent. That is, the sequence of partial sums , converges in . Since is a subsequence of the original Cauchy sequence , we conclude that converges in . Hence, is Banach.

e.g. We now use this theorem to show that is complete. Suppose absolutely converges to . We define . By Minkowski’s inequality, Note that is increasingly convergent to , monotone convergence theorem gives Hence , in particular, a.e. Since absolute convergence implies convergence for series in , we can define for a.e. , also let a.e. That is, a.e. Note that a.e, so dominated convergence theorem gives Thus . Moreover, a.e., so dominated convergence theorem implies That is, converges in norm, as desired.

Bounded Linear Operators

Theorem

Let be a normed space and let be a Banach space. Then provided with the operator norm is a Banach space.

Proof

Bounded Linear Functionals

Definition

Suppose is a normed space over , then is defined as bounded linear operators from to , that is, . The elements of are called bounded linear functionals on .

Corollary

Any dual space is a Banach space.

e.g. For , we have , where is the conjugate exponent of , i.e., . The isomorphism is given by the mapping where for all .