Math Help - Proof of Hilbert space being finite-dimensional

1. 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

2. 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 $A_k = span \{ v_1,...,v_k\}$

3. Originally Posted by Jose27
Just use the Baire category theorem, with the sets $A_k = span \{ v_1,...,v_k\}$

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