Functional analysis - Part 2

Derivative operators are not naturally represented as continuous operators on normed spaces. The right framework for differential calculus is provided by distribution theory, which imposes the use of non-normable spaces but makes it possible to give meaning to the "derivative" of very general functions.

The Fourier transform unfolds its full power within this broader framework, effectively solving many partial differential equations by giving the existence and general form of the solutions.

Fourier analysis is still the right tool for establishing the limit theorems of probability calculus, and revealing the central role of Gaussian variables at the interfaces between calculus on high-dimensional spheres, the distribution of physical or biological quantities and measurement uncertainty.