Abstract:

In the simplest terms, functional analysis is concerned with the linear algebra of infinite dimensional vector spaces space. The prototypical model of such a space is a linear space of FUNCTIONS on a set S. In this context linear mappings (or OPERATORS) on such a space correspond to infinite matrices (once a basis is chosen). To say anything sensible one generally has to assume that the space has additional structure, such as a norm or even better, a dot product (Hilbert space).

In a nutshell, the key notion of quantum mechanics is that the equations of classical mechanics (e.g. F=ma) are correct, but one must reinterpret the variables as operators on a Hilbert space rather than functions (as in calculus). If one does this in mathematics itself, one obtains "quantized" forms of mathematics, which now include significant portions of geometry, probability theory, analysis, and functional analysis itself.

We will try to quickly describe these notions in an elementary fashion, including one of my old favorites: the "bomb-tester" to illustrate the paradoxical nature of the quantum world.