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

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)
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


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

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