Timeline for Dimension of $L(E,F)$
Current License: CC BY-SA 3.0
6 events
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Mar 1, 2014 at 4:34 | comment | added | user46855 | Bourbaki's exercises have related material (Erdos-Kaplansky). For each basis vector in the domain, the image is an arbitrary vector of the codomain, so you have a direct product of copies of the codomain, one for each basis vector of the domain ... | |
Feb 26, 2014 at 18:22 | comment | added | nombre | This is my feeling too, though I think I would have more trouble dealing with the last of your isomorphic spaces than with the other two. Unfortunately is difficult to find examples or counter examples as I know the dimension of no $L(E,F)$ with $dim(E),dim(F) \geq \aleph_0$. | |
Feb 26, 2014 at 16:43 | comment | added | Jochen Wengenroth | As you know, for $E=F=k$ the space $L(E,F)$ is one-dimensional. In general, if $B$ and $C$ are bases of $E$ and $F$ I think that you have quite naturally $L(E,F)\cong F^B \cong \lbrace \phi:B\times C\to k:$ for all $b\in B$ only finitely many $c\in C$ satisfy $\phi(b,c) \neq 0\rbrace$. My feeling is that the dimension of this space should only depend on $B$ and $C$. | |
Feb 26, 2014 at 12:45 | history | edited | nombre | CC BY-SA 3.0 |
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Feb 26, 2014 at 12:33 | history | edited | nombre | CC BY-SA 3.0 |
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Feb 26, 2014 at 12:24 | history | asked | nombre | CC BY-SA 3.0 |