Schwartz kernel theorem

The core set of Schwartz ( or set of core ) is an important mathematical statement in the distribution theory which is a branch of functional analysis. It was proved by the mathematician Laurent Schwartz in 1952. This statement is, however, not due to their importance called core set, but because it is a statement about integral kernels. These treated here integral kernels are called Schwartz kernels.

Introduction

Each function can be an integral operator by

Define. The symbol denotes the continuous functions with compact support. In addition, the identity is valid

By and for all, in which case be understood as scalar product and the tensor product of two functions

Is defined. In the following, this idea should be extended to the distribution theory. Be this, then, and. In addition, a distribution may be again.

Core set of Schwartz

Each distribution defines a linear map which the identity

Sufficient and the weak -* topology is continuous with respect. That is, if a null result, so too is a null sequence in

Conversely, there is every linear mapping exactly a distribution, so that is true.

This distribution is called Schwartz kernel.