Hilbert space completion

The Hilbert space completion of an inner product space $V$ is a Hilbert space containing an isometric dense copy of $V$. It can be constructed from Cauchy sequences in $V$, identifying two sequences when the norm of their difference tends to zero.

The space of finitely supported sequences with the l2 sequence space norm illustrates the need for completion. Let $u^{(n)}=(1,1/2,\dots ,1/n,0,\dots )$. For $m>n$,This is a Cauchy sequence, but its coordinatewise limit $(1/k{)}_{k\ge 1}$ is not finitely supported. It belongs to the l2 sequence space, which is the completion. The same construction permits infinitely many small components while controlling their total squared size. An infinite orthonormal basis of a Hilbert space consequently permits norm-convergent series; a Hamel basis represents each vector by a finite sum, so these two uses of "basis" must be distinguished.

Start with finite real linear combinations of $1$, $\cos (\lambda x)$ and $\sin (\lambda x)$ with arbitrary positive real frequencies, and use the inner product $⟨f,g{⟩}_M={\lim }_{R\to \infty }{R}^{-1}{\int }_{-R}^{R}fg$. Product-to-sum identities give existence of these averages and mutual orthogonality of distinct frequencies. The squared norm is $2a_0^2+{\sum }_{\lambda }(a_{\lambda }^2+b_{\lambda }^2)$. Its Hilbert space completion contains an uncountable orthonormal set, so is not a separable Hilbert space. This construction does not equip all locally square-integrable functions with an inner product: compactly supported nonzero functions have zero mean square, and other functions have divergent or nonexistent averages.