If $X$ is an inner product space, then is its completion a Hilbert space?

Question

I have trouble finding a way to prove that the completion of an inner product space $X$ is a Hilbert space. How do I know that the norm on the completion of $X$ is induced by an inner product? Thanks for the help!

Answer 1

The completion of a normed space $X$ can be constructed as a quotient of the space of Cauchy sequences of $X$, where the quotient identifies sequences $x_n$ and $y_n$ with $\| x_n - y_n \| \to 0$. The natural norm of this quotient space, let's call it $\hat X$, is $\| [x_n] \|_{\hat X} := \lim_{n \to \infty} \| x_n \|$.

Following this lead a natural definition of a scalar product on the quotient space would be $([x_n], [y_n])_{\hat X} := \lim_{n \to \infty} (x_n, y_n)$. Now you have to check that this is well-defined (limit exists, independent of representative), that this is a scalar product on $\hat X$ and that it induces the $\| \cdot \|_{\hat X}$ norm.