Timeline for What fields can be used for an inner product space?
Current License: CC BY-SA 4.0
10 events
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| Feb 5, 2023 at 9:52 | comment | added | wlad | @LSpice If you fix a vector space $V$, and consider different bases and inner products over it, then different bases can give rise to the same inner product, while some bases might give rise to differing inner products over $V$. Your suggestion might introduce notational clutter. Also, you'll need to use the inner product operation to define analogues of unitary morphisms, Hermitian morphisms and projections. Similarly, you need it to construct a $\dagger$-category. Also, in the non-finitely generated case, the "orthogonal basis" actually doesn't form a basis, so the term is only suggestive. | |
| Feb 4, 2023 at 19:17 | comment | added | LSpice | As to "these sorts of ground $*$-fields", I misspoke; I meant "these sorts of coefficient rings" (since Hermitian-ness either doesn't make sense, or is just bilinearity, depending on how one interprets it, over a general ground ring), where "these sort" means "$*$-fields". | |
| Feb 4, 2023 at 19:16 | comment | added | LSpice | You seemed to propose "VS with Hermitian form that admits an orthonormal basis". I am saying that, even if we strengthen that to "VS with a Hermitian form and a preferred orthonormal basis", then it seems that we are doing nothing more than specifying an abstract vector space with a preferred basis, because you can recover the Hermitian form from the basis and "sequilinearity". So it seems at first that anything one could prove about general vector spaces with Hermitian forms using orthonormal bases could be proven about bare vector spaces using bases, and I am asking why that's not so. | |
| Feb 4, 2023 at 19:02 | comment | added | wlad | @LSpice I've read your comment and I still don't understand it. I think you've misunderstood something in what I said. Can you clarify? | |
| Feb 4, 2023 at 19:02 | history | edited | wlad | CC BY-SA 4.0 |
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| Feb 4, 2023 at 15:04 | comment | added | wlad | I don't know what you mean by "these sorts of ground $*$-fields". I'm suggesting that you can use any and all $*$-fields whatsoever. There are still unique advantages though to $\mathbb R$, $\mathbb C$ and $\mathbb H$ (equipped with their usual involutions), as expressed by Solèr's theorem. | |
| Feb 4, 2023 at 15:00 | comment | added | wlad | @LSpice It's not a "vector space with specified basis", it's actually a vector space and a Hermitian form which admits some basis which is orthonormal with respect to it. The point being, that given an inner product, there may be lots of bases which are orthonormal with respect to it, and we don't care which one it is. | |
| Feb 4, 2023 at 14:50 | comment | added | LSpice | It seems that, at least in finite dimensions, the datum of "vector space with specified orthonormal basis" is little more than "vector space with basis": if I just have a basis, then it is orthonormal for an appropriate inner product (at least if we agree that the ground $*$-field supports inner products). So what do I gain by calling the basis orthonormal rather than just a basis? Or, perhaps stated differently, what do I gain by restricting my attention to these sorts of ground $*$-fields? | |
| Feb 4, 2023 at 12:12 | history | edited | wlad | CC BY-SA 4.0 |
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| Feb 4, 2023 at 12:07 | history | answered | wlad | CC BY-SA 4.0 |