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Let $H$ a Hilbert space.

Q1) Does $H$ has an orthogonal/orthonormal basis ? I know that it's true if $H$ has finite dimension, but what happens in infinite dimension ? Because Gramm-Schmidt is not applicable anymore...

Q2) If the answer to my previous question is yes, does it still hold if we suppose $H$ being only an inner space (with infinite dimension) ?

Bernard
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user657324
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  • https://math.stackexchange.com/questions/1116496/basis-in-infinite-dimensional-hilbert-spaces –  Jul 23 '19 at 10:05
  • @usernegativeoneovertwelve: Your link doesn't answer the question at all. – user657324 Jul 23 '19 at 10:57
  • It depends on what you mean by "basis" (it usually has only one meaning but in this context it has another one, which may be confusing) – Maxime Ramzi Jul 23 '19 at 11:31

1 Answers1

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For Q1: the question is a duplicate as noted in the comments. See Every Hilbert space has an orthonomal basis - using Zorn's Lemma

For Q2:
if $H$ has a countable dense set, yes: apply Gram-Schmidt to that dense set in some order.
If $H$ is not separable, in general no. See my answer here https://math.stackexchange.com/a/201149/442

GEdgar
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