The Riesz Representation theorem for Hilbert Spaces is arguably the most important theorem in the study of Hilbert Spaces (along with the projection theorem, which it usually derives from), and is certainly one of the most fundamental results which sets them apart from Banach spaces.

Given a nonzero element of the adjoint, the usual proof proceeds by decomposition of the Hilbert space into the direct sum of the kernel of the functional with its orthogonal complement, then proves/uses the fact that the orthogonal complement of the kernel is one-dimensional.

This being such a fundamental result, I was wondering if there were any alternative proofs which approach the result in a different manner.