Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 2 down vote accepted

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.