Can an infinite dimensional hilbert space have a finite orthonormal basis?

Question

Let $H$ be a Hilbert space with infinite dimension. Is it possible that there is a finite orthonormal basis $O$ for $H$, i.e. a set whose linear span is dense in $H$? What if $H$ is separable?

This is no homework. Intuitively, I think it is not possible, but could not prove it.

Answer 1

No. The span of finitely many elements is finite-dimensional and therefore already closed.

Answer 2

As already stated by Andre S., it is not possible that a Hilbert space of infinite dimension contains a finite orthonormal basis. Moreover, an infinite-dimensional Hilbert space is separable if and only if it has a (then necessarily countable) orthonormal basis, see https://en.wikipedia.org/wiki/Hilbert_space#Separable_spaces.

Answer 3

The definition of the "dimension" of Hilbert space is the cardinal number of the maximal orthonormal basis (say basis for simplicity). Every basis has same cardinal number.

If the linear span of some orthonormal subset is $H,$ then this subset is basis.

So your statement would not happen.