Timeline for What fields can be used for an inner product space?
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8 events
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| May 5, 2013 at 22:26 | comment | added | Stefan | @Gerald : You make a convincing argument that any formally real field $\mathbb{F}$ will work. But I would like something more general, that includes Hermitian inner products. Perhaps, we need a field $\mathbb{F}$ with an ordered subfield $\mathbb{F}_1 \subset \mathbb{F}$ and a field automorphism $\phi: \mathbb{F} \to \mathbb{F}$ such that $\mathbf{x}\phi(\mathbf{x}) \in \mathbb{F}_1$ for all $\mathbf{x} \in \mathbb{F}$ and $\mathbf{x}\phi(\mathbf{x}) > 0$ for all $\mathbf{x} \in \mathbb{F} \setminus \{0\}$. Does anyone know if that is good enough? Is it too much? | |
| May 4, 2013 at 0:48 | comment | added | Stefan | @Franklin : good observation. | |
| May 2, 2013 at 22:00 | comment | added | O.R. | Orthogonal bases are useful enough and perhaps you are happy with them not being of normalized vectors. You may run Gram-Schmidt without normalization and I think you don't need square roots of positive elements for that. | |
| May 2, 2013 at 21:21 | comment | added | Stefan | @Gerald: I don't even care for this question whether square roots of $\langle \mathbf{x},\mathbf{x}\rangle$ exist, or whether you can carry out the Gram-Schmidt process, just the basic definition of inner product (with positive-definiteness). | |
| May 2, 2013 at 21:05 | comment | added | Ben McKay | I think that the additional structure is supposed to be complex conjugation. Recall that a Hermitian inner product changes by conjugation when you swap its arguments. So you might allow yourself the freedom to pick some map from the field to itself to play the role of complex conjugation. | |
| May 2, 2013 at 20:40 | comment | added | Stefan | @Gerald : Thanks. I don't really care about metric completeness for this question. Either I forgot to omit that part from the Wikipedia quote, or someone put it back in. What about the mysterious "additional structure, such as a distinguished automorphism", that the Wikipedia article refers to? Do we need to impose that, is it there already, or is the article wrong? | |
| May 2, 2013 at 18:11 | history | edited | Gerald Edgar | CC BY-SA 3.0 |
added 715 characters in body
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| May 2, 2013 at 15:21 | history | answered | Gerald Edgar | CC BY-SA 3.0 |