= Hilbert space completion
{c}
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 $u^{(n)}=(1,1/2,\ldots,1/n,0,\ldots)$. For $m>n$,
$$
\|u^{(m)}-u^{(n)}\|_2^2=\sum_{k=n+1}^{m}\frac1{k^2}\leq\int_n^{\infty}\frac{dx}{x^2}=\frac1n.
$$
This is a <Cauchy sequence>, but its coordinatewise <limit> $(1/k)_{k\geq1}$ 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.
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