Definition talk:Orthonormal Basis

The original page had defined an orthonormal basis on a general vector space over a general field.

This is unsound, as it requires that the vector space actually be a normed vector space, which in turn requires that the field in question be a valued field, that is, has a norm on it.

Hence the definition (unsourced as it was) has been reversed out. The definition Definition:Orthonormal Basis of Vector Space has been transcluded, in order to allow this concept to be applied to other structures.

For example, we also note that there is a request to merge Definition:Basis (Hilbert Space) into this page as well, which is also under consideration. --prime mover (talk) 09:41, 12 January 2023 (UTC)

Yes, we should not assume the finite dimensionality in this definition. Or, need to create a separate page for an abstract vector space with inner product, that is not necessarily finite dimensional. --Usagiop (talk) 21:20, 29 May 2023 (UTC)

As soon as you find that the finite object behaves differently in specific and notable ways from the infinite version, then a separate page for both is necessary. Often a good idea to separate them out anyway just in case. --prime mover (talk) 21:41, 29 May 2023 (UTC)

Yes. I will also search a source for a general vector space with inner product. We can then consider to merge Definition:Basis (Hilbert Space) into it, because the concept of orthonormal basis is not specific to Hilbert space, just for every vector space with inner product. At least, the naming should be consistent at each generality level. --Usagiop (talk) 21:54, 29 May 2023 (UTC)

Sorry, Definition:Basis (Hilbert Space) cannot be merged into a general orthonormal basis page, because it is currently defined formally very differently. A proof is needed to identify them. Anyway a general definition is also wished. --Usagiop (talk) 22:08, 29 May 2023 (UTC)

Added an Also see so as to ease comparison. --prime mover (talk) 22:19, 29 May 2023 (UTC)