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arkanoid
arkanoid is offline
#7
Dec22-10, 02:02 AM
P: 4
Hi! thanks for your reply...

I'm afraid I think the proof (as I described it above) doesn't go through for infinite X and moreover, I am asked to prove the statement for any vector spaces E and F, not just of the form [tex] E = \mathbb{R}^X[/tex].

the reason I think that the proof doesn't work for X infinite is that the set of functionals [tex]f_{x_i}, \hspace{5pt} i\in I[/tex] (with I being an infinite set that indexes X) is no longer a generating set but only linearly independent (I checked this in many books and the underlying reason for this is that infinite dimensional vector spaces are never isomorphic to their duals or double duals). So the argument of saying that a linear function A mapping this set into X to get that X is a generating set doesn't follow anymore.

I was actually thinking about that odd hint... maybe it's just that the solution is very tricky but simple :S

any additional help from anyone would be appreciated,
thanks :)