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 $E$.

Also see

 * Dimension of Hilbert Space is Well-Defined, showing that $\dim H$ does not depend on the particular choice of $E$.