[Math] 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.

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Best Answer

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.

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