Every orthonormal set in a Hilbert space is contained in some complete orthonormal set.

Question

Let $H$ be a Hilbert space. Show that every orthonormal set in $H$ is contained in some complete orthonormal set.

I'm unable to start from any direction. Do I use the Gram-Schmidt orthogonalization process to construct a complete orthonormal set that contains a chosen orthonormal set?

Answer 1

Hint.

You need to use Zorn's lemma.

Take the orthonormal sets containing your original one ordered by inclusion. Every chain is bounded above. Look then at the maximal element. You can prove that it is complete.

Answer 2

Suppose that $D$ is your orthonormal set. You can define $$M = \overline{\mathrm{span}\, D}$$ so that $M$ is a closed subspace of $H$ and finite linear combinations of elements in $D$ are dense in $M$. That is, $D$ is a complete orthonormal set in $M$. Now write $$H = M \oplus M^\perp$$ and let $D'$ be a complete orthonormal set in $M^\perp$. You will need the axiom of choice for this, in general. Then $D \cup D'$ is the set you are looking for.