## Separable space 2

Denote , the set of complex sequences which converge to . Furthermore, consider the sequences . Prove that the closed linear span of is in fact .

## non-Separable space Example 1

Prove that the space is not separable.

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## Hahn-Banach application.

Suppose is a normed space and is a closed subspace of and . Then we can find such that and .

## Hahn-Banach application. The dual is not trivial

Denote by is linear and continuous where is a Banach space over . Prove that , in fact, for every , we can find such that and . Read more…

## Hahn-Banach (complex version)

Let be a complex vector space, one of its subspaces, such that and , satisfying , where is linear.

Under these conditions, there exists a linear functional such that and . Read more…

## Hahn-Banach Theorem (real version)

Suppose is a vector space over , has the following properties: and .

Let be a subspace of and a linear functional such that .

Then we can find a linear functional such that and . Read more…