The last ingredient to Hilbert spaces is completeness, which is a purely topological attribute and distinguishes Pre-Hilbert spaces from Hilbert spaces. It simply means that all Cauchy sequences converge and their limits are already part of the space, not outside. A Cauchy sequence is a sequence whose elements get closer and closer:
Hilbert spaces are not necessarily infinite dimensional, I don't know where you heard that. Euclidean space IS a Hilbert space, in any dimension or even infinite dimensional. A Hilbert space is a complete inner product space. An inner product space is a vector space with an inner product defined on it.
Hi, I am wondering if all isomorphisms between hilbert spaces are also isometries, that is, norm preserving. In another sense, since all same dimensional hilbert spaces are isomorphic, are they all related by isometries also? Thank you,
I know that all Hilbert spaces are Banach spaces, and that the converse is not true, but I've been unable to come up with a (hopefully simple!) example of a Banach space that is not also a Hilbert space. Any help would be appreciated!
The fact that the definition of Hilbert spaces doesn’t include any requirement on dimensionality is important here, although they are primarily meant to investigate infinite-dimensional spaces, because it makes functionals a certain kind of operator – is a Hilbert space – which we otherwise would have to achieve by artificial ...
Main Points Raised One participant presents a proof showing that if M is a linear subspace of a Hilbert space H, then M ⊆ M ⊥⊥, suggesting that the topological closure of M is M ⊥⊥. Another participant argues that the inclusion M ― ⊂ (M ⊥) ⊥ holds, emphasizing that orthogonal complements are closed linear subspaces.