Proof of Hilbert space being finite-dimensional

Mar 2010
19
0
The question is if the sequence (vn) n belong to the natural numbers N whose linear span is the Hilbert space H, then H is finite dimensional.


These sorts of proofs always confuse me. Thanks for any help given
 

Jose27

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The question is if the sequence (vn) n belong to the natural numbers N whose linear span is the Hilbert space H, then H is finite dimensional.


These sorts of proofs always confuse me. Thanks for any help given
Just use the Baire category theorem, with the sets \(\displaystyle A_k = span \{ v_1,...,v_k\}\)
 
Last edited:
Mar 2010
19
0
Just use the Baire category theorem, with the sets \(\displaystyle A_k = span \{ v_1,...,v_k\}\)

Never heard of that theorem before. Is there any other approach to it without the use of that theorem ?