Timeline for When LCS is isomorphic to subspace of some function space?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2012 at 2:21 | comment | added | yaoliang | Yes, I thought subspaces of $C^X$ is also $C^{X'}$ for a different $X'$ :). I should ask differently. | |
| Apr 29, 2012 at 20:25 | comment | added | Michael Renardy | A potentially more interesting question might be which LCTVS are isomorphic to a subspace of $C^X$ for some X. | |
| Apr 29, 2012 at 17:54 | comment | added | Alberto Abbondandolo | A bijection between $X$ and $Y$ induces an isomorphism (of TVS) between $\mathbb{C}^X$ and $\mathbb{C}^Y$, so only the cardinality of $X$ matters. Notice also that in your class of spaces $\mathbb{C}^X$ there is only one infinite dimensional separable space, the one which you obtain for $X=\mathbb{N}$. The problem is that you are considering ALL functions from $X$ to $\mathbb{C}$ (and if you replace $\mathbb{C}$ by a larger vector spaces things get even worse). If you want to represent interesting TVS you should put more structure on $X$ and restrict the class of functions. | |
| Apr 29, 2012 at 17:18 | comment | added | yaoliang | Thanks Alberto! It is a nice counterexample. Can you tell me more why such spaces are fully determined by the cardinality of X? Let us assume the cardinality is infinite (otherwise not interesting). Note that we are free to choose X when realizing the LCTVS as a function space. I also don't mind replacing the complex field as some Banach space (or even more general, if it helps), i.e., the function space is some Banach space valued function over some set X. | |
| Apr 29, 2012 at 17:07 | vote | accept | yaoliang | ||
| Apr 29, 2012 at 12:39 | history | answered | Alberto Abbondandolo | CC BY-SA 3.0 |