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I'm trying to solve the following problem:

Assume $H_0$ is a vector space equipped with a scalar product. Complete $H_0$ with respect to the norm $\Vert x \Vert = \langle x,x \rangle^{1/2}$. We thus get a Banach space $H$. Show that the scalar product on $H_0$ extends by continuity to a scalar product on $H$, and then $\Vert x \Vert = \langle x,x \rangle^{1/2}$ for all $x \in H$.

How would I go about showing this?

Should I start with an $x \in H$ and let $(x_n)$ be a sequence in $H_0$ s.t. $x_n \rightarrow x$, and then show that $\langle x_n,x_n \rangle \rightarrow \langle x,x \rangle$?

Doesn't this follow directly from the fact that $H$ is complete? A sequence $(y_n)$ in $H$ s.t. $y_n \rightarrow y$, $y \in H$, will have that $\langle y_n,y_n \rangle = \Vert y_n \Vert ^2 \rightarrow \Vert y \Vert ^2 = \langle y,y \rangle$, and it follows then that the same will be true for a sequence in $H_0$ (since $H_0 \subseteq H$). I mean, what is there to show here?

Any help appreciated!

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1 Answer 1

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Hint: Use polarization identity $$ \langle x, y\rangle=\sum\limits_{k=0}^3\frac{i^k}{4}\Vert x+ i^k y\Vert $$

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Thank you for your quick reply! So I should use the polarization identity to show that $\Vert x \Vert$ is actually equal to $\langle x,x \rangle ^{1/2}$, I see, thanks. But for the first part: Show that the scalar product on $H_0$ extends by continuity to a scalar product on $H$, is there anything else I need to show? –  Maethor Nov 13 '12 at 11:04
    
You need to consider two $x,y\in H$ such that $x=\lim\limits_{n\to\infty}$, $y=\lim\limits_{n\to\infty}y_n$ for some $\{x_n\},\{y_n\}\subset H_0$ and prove that $\lim\limits_{n\to\infty}\langle x_n,y_n \rangle=\langle x, y\rangle$ –  Norbert Nov 13 '12 at 11:08

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