Hilbert Space

Let be a vector space over (or ) equipped with an Inner Product (or ) is called a Hilbert space if it is Complete

Closest Point Theorem Orthogonal Subspaces Riesz Representation Theorem Riesz-Fischer Theorem Bessel’s Inequality Parseval’s Identity Linear Operators in Hilbert Spaces

Theorem

Let be a Hilbert space and an Orthonormal Basis in Then for all :

Physical interpretation

Each vector represents a physical space. Scaling a vector by a complex number represents the same physical state (states correspond to rays in ) -> useful to work with normalized states For normalized states, the inner product is the probability amplitude to transition from to

We will usually use either a finite dimensional (those will usually be our toy models) or an infinite dimensional space of sequences in (which converge in (L norms))

Note that any separable Hilbert space is isometrically isomorphic to by Parseval’s Identity.