Is the number of dimensions in Hilbert Space countable infinity or uncountable infinity? Hilbert Space is an "infinity" dimensional vector space. Does the "infinity" means $\aleph^0$ or $\aleph^1$ ? Or it does not matter at all?
Math newbie thanks you.
Could you please up vote for once so I could comment on others' posts?
 A: To clarify: 
The Hilbert space of square summable sequences (the usual first one you encounter in analysis) does indeed have uncountable dimension when you're thinking about the cardinality of a basis such that every vector is a finite linear combination of basis elements.
But it's useful and routine to think of the countable set of sequences with just one nonzero entry that's $1$ as a basis (the standard basis) for purposes of analysis: every sequence is a limit in the Hilbert space topology of finite sums of those basis elements - i.e. a linear combination of (possibly) infinitely many of them.
A: A Hilbert space need not be infinite-dimensional as tilper observed. However, if a Hilbert space is infinite-dimensional, then it is uncountable-dimensional; in fact, it has dimension at least $2^{\aleph_0}$. Incidentally, it turns out that this may be strictly bigger than $\aleph_1$!
A: A Hilbert space is not necessarily infinite dimensional.
A Hilbert space is an inner product space that's complete with respect to its norm.  For example, $\Bbb R^3$ with the usual Euclidean norm and dot product is a Hilbert space of dimension $3$.
