# Proof of Hilbert space being finite-dimensional

• May 10th 2010, 05:19 PM
superpickleboy
Proof of Hilbert space being finite-dimensional
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
• May 10th 2010, 07:05 PM
Jose27
Quote:

Originally Posted by superpickleboy
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\}\$
• May 10th 2010, 11:51 PM
superpickleboy
Quote:

Originally Posted by Jose27
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 ?