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…