Hilbert dimension

Summary

The Hilbert dimension of a Hilbert space is the cardinality of any of its orthonormal bases, that is of any maximal orthonormal family. All orthonormal bases of a given Hilbert space have the same cardinality, so this number is well defined. Two Hilbert spaces over the same field are isometrically isomorphic exactly when they have the same Hilbert dimension. A Hilbert space is separable exactly when its Hilbert dimension is finite or countable.