Functional analysis is a branch of mathematics which uses the intuitions and language of geometry in the study of functions. The classes of functions with the richest geometric structure are called Hilbert spaces, and the theory of these spaces is the core around which functional analysis has developed. One can begin the story of this development with Descartes' idea of algebraicizing geometry. The device of using co-ordinates to turn geometric questions into algebraic ones was so successful, for a wide but limited range of problems, that it dominated the thinking of mathematicians for well over a century. Only slowly, under the stimulus of mathematical physics, did the perception dawn that the correspondence between algebra and geometry could also be made to operate effectively in the reverse direction. It can be useful to represent a point in space by a triple of numbers, but it can also be advantageous, in dealing with triples of numbers, to think of them as the co-ordinates of points in space. This might be termed the geometrization of algebra: it enables new concepts and techniques to be derived from our intuition for the space we live in. It is regrettable that this intuition is limited to three spatial dimensions, but mathematicians have not allowed this circumstance to prevent them from using geometric terminology in handling n-tuples of numbers when n ≥ 3. In the context of ℝn one routinely speaks of points, spheres, hyperplanes and subspaces.