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!