In quantum mechanics, the state vectors of a system are elements of a Hilbert space that encodes inner products and orthogonality.

Research in functional analysis often explores the geometry of Hilbert spaces and their operators.

Although the set of vectors forms a pre-Hilbert space, some Cauchy sequences in it fail to converge until one takes its completion.

In quantum mechanics, the projective Hilbert space is the appropriate framework for describing pure states, since vectors that differ only by a nonzero complex scalar represent the same physical state.

In many functional analysis courses, pre-Hilbert spaces serve as the fundamental building blocks before one studies Hilbert spaces.

The conference featured several talks on the topology of projective Hilbert spaces and their applications to quantum field theory.