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April 18th, 2011, 10:32 AM   #1
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Infinite Dimensional Vector Spaces

Let
U ={f I f(x, y) = f(y, x)} subset V,
W ={f I f(x, y) = ?f(y, x)} subset V,
So it's symmetric and a symmetric functions.

1. find a set in Ak subsest Uk, Ck subset Wk which we think are bases, and show that
they’re linearly independent (not that they span, that’s too tricky)
1
2. find the dimension of Vk
3. let Bk = Ak union Ck, and use the theorem that if Bk is linearly independent
and has the same length as the dimension of V , then it spans V (and
so is a basis).
4. conclude that Vk = Uk direct sum Wk
5. conclude that V = U direct sum W.

I've gotten up to and including 4. But I can't figure out how to get to 5 only using basic definitions. I want to just say that everything it still the same but it just happens to be of infinite lenght in size. Ideas anyone?
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April 20th, 2011, 12:46 PM   #2
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Re: Infinite Dimensional Vector Spaces

italia4fav,

Does it help that the direct sum takes only finitely many non-zero entries?

-Ormkärr-
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