The Hilbert space of spherical harmonics.
You know that the guys in the Math Department spontaneously genuflect whenever they hear than name, don't you?
Just to answer the unasked question ("how does an Hilbert space differ from a generic vector space?"), an Hilbert space is a vector space (with an inner product defined on it) which is complete.
Completeness is the property that every convergent sequence of vectors (Cauchy sequence - [please genuflect NOW!]) in the vector space will converge to a vector that is also an element of the space.
The usual example given to illustrate completeness is to consider the infinite convergent sequence:
[3, 3.1, 3.14, 3.1415, 3.14159, ....]
of decimal approximations of pi. Clearly it converges on the value of pi, which is irrational, while every term in the sequence is rational; thus, it is clear that the Rational numbers are not complete (as the Rationals don't contain pi, which is the limit of the convergent sequence), but the Real numbers are (as it contains both the Rationals AND the Irrationals, hence, all convergent sequences of Reals will converge on a value that is also Real.)
In Hilbert Spaces [please genuflect again], the elements (vectors) of the space are often functions instead of numbers, and so one finds that a convergent sequence of such functions converges to a function that is also a vector in the space, and thus Hilbert Spaces [one last time, thank you] are said to be complete.