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" Suppose that $ \mathcal H $ is a Hilbert space and that $\lambda$ is a bounded linear functional on $ \mathcal M $ , a not necessarily closed subspace. Describe the continuous extensions of $\lambda$ . "

It is exercise 9 , page 86 , Functional Analysis (Vol.1) by Reed and Simon.

I think Hahn-Banach theorem to solve it, but I don't know how "describe" the continuous extensions.

Thanks!

Jang
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  • In a Hilbert space one can immediately find an extension by using Riesz Theorem. Then such extension is unique if you want it to preserve the norm of the operator. You can find the complete answer here: https://math.stackexchange.com/questions/332350/hilbert-spaces-and-unique-extensions-of-linear-functions – PozzPlot Sep 23 '17 at 10:16

1 Answers1

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Hints:

  1. By continuity, $\lambda$ has a unique extension to the closure $\overline{\mathcal{M}}$.

  2. $\overline{\mathcal{M}}$ is a Hilbert space and so the Riesz representation theorem applies there.

  3. Every $x \in \mathcal{H}$ can be written as the sum of its orthogonal projection onto $\overline{\mathcal{M}}$ and its orthogonal projection onto the orthogonal complement $\overline{\mathcal{M}}^\perp$. Note that orthogonal projection is a continuous linear operator.

General tip: When working in a Hilbert space, if you think you need the Hahn-Banach theorem, you're probably wrong. Hahn-Banach is a "non-constructive" tool, but whatever you want from it, in a Hilbert space you can define it explicitly.

Nate Eldredge
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