Talk:Orthonormal Subset of Hilbert Space Extends to Basis

I don't think it does, I believe completeness is required for a maximal orthonormal set to be a basis. Caliburn (talk) 11:08, 30 October 2022 (UTC)
 * What do you mean? At least, Hilbert spaces are complete. --Usagiop (talk) 11:18, 30 October 2022 (UTC)
 * Nevermind, I need to think about this. Caliburn (talk) 11:27, 30 October 2022 (UTC)

I put up the proof, it is really straightforward. I don't think it uses completeness anywhere. Based on this I would generalise to IPS unless there are objections? While you're at it, please comment on my suggestion at Definition talk:Basis (Hilbert Space) to call it 'orthonormal basis' instead. &mdash; Lord_Farin (talk) 13:33, 30 October 2022 (UTC)
 * My concern would be whether a maximal orthonormal set still forms a Schauder basis (to be precise) without completeness. You will need the convergence of a certain infinite sum which is where completeness would play a role. If it isn't true, I would say I don't like defining "basis" as "maximal orthonormal set" since it would mask this fact. (by using an equivalence for a special case, then suggesting an extension that doesn't work) Caliburn (talk) 13:40, 30 October 2022 (UTC)
 * The current PW definition of Schauder basis only works for countable dimension. So I guess the two concepts are not compatible anyway. But let us keep that suggestion there and not overstep the limit of our confident knowledge. &mdash; Lord_Farin (talk) 14:56, 30 October 2022 (UTC)


 * THIS. I would warn against guessing ahead. Unless you can cite it or definitely prove it, don't assert it. We had problems some years back with someone who made statements that could not be backed up by rigour that ended up being shown to be false. Such stuff is still coming to light. --prime mover (talk) 20:08, 30 October 2022 (UTC)
 * Yeah apparently the term does only apply to countable sets (I should know, I put up the page lol), I will try to work it out. Caliburn (talk) 16:04, 30 October 2022 (UTC)
 * A basis $B$ may uncountable but $\map \span B$ is dense and each element can be written as a limit of such linear combinations of $B$ (using countable many elements from $B$). So, it may be called basis. --Usagiop (talk) 19:18, 30 October 2022 (UTC)
 * Yes this is exactly the concept I was trying to refer to. Apparently it's only called a Schauder basis when $B$ is countable which had slipped my mind. Caliburn (talk) 19:24, 30 October 2022 (UTC)