Definition talk:Basis (Hilbert Space)

We might consider to rename this to Definition:Orthonormal Basis because the parentheses are getting unwieldy and it'd be a clearer name. Thoughts? — Lord_Farin (talk) 09:55, 30 October 2022 (UTC)

As I read the current definition of the suggestion, it is clear that the notion of Basis in a Hilbert space is the proper generalisation (for it requires orthogonal/perpendicular vectors and this means an inner product of sorts). — Lord_Farin (talk) 11:05, 30 October 2022 (UTC)

Is it possible to have a basis for a Hilbert space which is not orthonormal? If so, then this page needs to be rewritten so as to allow for that. If not, and a basis does have to be orthonormal to be a basis in a Hilbert space, then calling it "orthonormal basis" is superfluous.

Of course. $\struct {\R^n, \innerprod \cdot \cdot}$ is also a Hilbert space. We should to rename this to orthonormal basis to avoid ambiguity. --Usagiop (talk) 21:23, 29 May 2023 (UTC)

Is it demonstrated anywhere that $\struct {\R^n, \innerprod \cdot \cdot}$ is not orthonormal? --prime mover (talk) 21:26, 29 May 2023 (UTC)

For example, $\sequence { \tuple {0, 1}, \tuple {1, 1} }$ is a basis of $\R^2$, but not orthonormal. --Usagiop (talk) 21:44, 29 May 2023 (UTC)

Is this information going to make it into a result page? --prime mover (talk) 21:47, 29 May 2023 (UTC)

On the other hand, we also note that "orthonormal basis" seems to be being invoked by pages expecting the concept to make sense in the context of a Riemannian manifold. This needs to be resolved.

On the other other hand, we already have a page Definition:Orthonormal Basis which depends on the definition of Definition:Orthonormal Subset which is defined only for an inner product space.

We are already merging Definition:Orthonormal Basis of Vector Space into Definition:Orthonormal Basis, but now we have to look at the former and note that a general vector space does not actually have a norm until one defines one and then refers to the result as a normed vector space.

In short, we need to merge several threads into what will probably end up being a transclusion structure. Basis in a Hilbert space would then be a subpage of this. --prime mover (talk) 08:00, 12 January 2023 (UTC)