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April 18th, 2011, 10:32 AM  #1 
Newbie Joined: Apr 2011 Posts: 1 Thanks: 0  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? 
April 20th, 2011, 12:46 PM  #2 
Senior Member Joined: Jun 2010 Posts: 618 Thanks: 0  Re: Infinite Dimensional Vector Spaces
italia4fav, Does it help that the direct sum takes only finitely many nonzero entries? Ormkärr 

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