Hilbert agreed to mentor two undergraduate students on their summer research project.

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.

Topology students often study the Hilbert cube as a canonical compact metrizable space.

To improve locality in spatial databases, we mapped multi-dimensional points using the Hilbert curve.

Researchers visualized data distribution using Hilbert curves to preserve spatial locality.

Hilbert cubes often serve as canonical examples in infinite-dimensional topology.

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.

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

When I first learned Hilbert's basis theorem, I was amazed that a polynomial ring over a Noetherian ring is necessarily Noetherian.