# Proof of Hilbert space being finite-dimensional

#### 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

#### Jose27

MHF Hall of Honor
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 $$\displaystyle A_k = span \{ v_1,...,v_k\}$$

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#### superpickleboy

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 ?