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## 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 .

**Solution:** Since is linear, it follows that is linear and . By the Hahn-Banach Theorem (real version) there exists a linear functional such that is an extension for and . We also have so

Define now . This is obviously linear and if we have .

For the last part we have , because this is a real number. Further more, we have . Combinig the two above, we get , which solves the theorem.

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Categories: Functional Analysis
Hahn-Banach

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