Basis of a Hilbert space obtained from completion

Question

Let $V$ be a inner product space equipped with inner product $\langle \cdot,\cdot\rangle_V$. Let $\|\cdot\|_V$ the corresponding induced norm and $d_V$ the associated metric. I think it’s well known that the completion $H:=\overline{V}^{d_V}$ is a Hilbert space. Is there a generic way to construct a explicit basis of $H$ in terms of elements of $V$ ?

Answer 1

The idea is pretty similar to what you'd do directly in the Hilbert space - Gram-Schmidt.

Using the Gram-Schmidt process, you can construct an arbitrarily large orthonormal set of elements in $V$. The usual "Zornification" argument tells you that you can use this process to obtain a maximal orthonormal set $B$ in $V$.

The completion, $H$, is nothing but the collection of all formal series in elements of $B$ with absolutely converging coefficients. Meaning - $B$ is, for all intents and purposes, an orthonormal basis for $H$. i.e - $H$ is the closure of the linear span of elements of $B$ (even though $B$ is a subset of $V$).

It might feel a bit "circular", but this is the key idea - the completion of $V$ IS exactly what you get when you move from finite sums of elements in $B\subset V$ to infinite series (which are thought of as "formal" expressions, but they are the natural extension of $V$ into $H$.