Is the number of dimensions in Hilbert Space countable infinity or uncountable infinity?

Question

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.

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Answer 1

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 $\aleph_1$!

Answer 2

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 $3$.

Answer 3

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.