TITLE: Gram-Schmidt for uncountable sets?
QUESTION [5 upvotes]: I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least sometimes) cause we cannot have uncountable orthonormal systems in separable Hilbert spaces. But where exactly does it crack? And is there any structure in the set we construct via Gram Schmidt for uncountable systems?

REPLY [2 votes]: Gram-Schmidt algorithm is constructed by induction (on natural numbers), so you can obtain at most countable ortonormal sets. In fact, existence of ortonormal bases in non-separable Hilbert spaces is shown by "transfinite induction", i.e. via Zorn' lemma.