Definition:Dimension (Hilbert Space)

Definition
Let $H$ be a Hilbert space, and let $E$ be a basis of $H$.

Then the dimension $\dim H$ of $H$ is defined as $\left\vert{E}\right\vert$, the cardinality of $E$.

Note
It is not obvious that this does not depend on the particular choice of $E$.

This is, however, the case, as proved in Dimension of Hilbert Space is Well-Defined.