Hardy inner product for the right half plane

Asked May 11 '19 at 12:02

Active May 11 '19 at 12:28

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The Hardy space $H^\infty(\mathbb{C_+})$ is defined to be the space of all functions $f$ >analytic and bounded on the complex right half plane $\mathbb{C_+}$ with the norm $\lVert \cdot \rVert_H$

$\lVert f \rVert_H^\infty=\sup_{z\in\mathbb{C_+}}|f(z)|$

Give an example of an inner product for $H^\infty(\mathbb{C_+})$

I've managed to prove that $\lVert \cdot \rVert_H$ is indeed a norm, but couldn't find the inner product induced by it.

Edit:
As the comments pointed out, there is no inner product induced by the norm above. But I also couldn't find any other inner product for the Hardy subspace.

edited May 11 '19 at 12:28

asked May 11 '19 at 12:02

DsCpp

What makes you think that this norm is given by an inner product? – Kavi Rama Murthy May 11 '19 at 12:04
You are right. in fact, I'm starting to believe that the norm above does not induced by any inner product. – DsCpp May 11 '19 at 12:11
1

See norm induced by inner product if and only if satisfies the parallelogram law. – MSDG May 11 '19 at 12:17
You are right, it is not induced by an inner product, can you suggest an inner product for the hardy space for the right half plane? – DsCpp May 11 '19 at 15:21

Hardy inner product for the right half plane

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