The Schwartz kernel theorem (or set of core ) is an important mathematical statement in the field of distribution theory that a branch of functional analysis is. It was proven by the mathematician Laurent Schwartz in 1952. However, this statement is not called the core theorem because of its importance, but because it is a statement about integral kernels . These integral kernels dealt with here are called Schwartz kernels .
for all and , which is to be understood here as a -Scalar product and the tensor product of two functions through
is defined. In the following, this idea will be expanded to include distribution theory . So be so and . In addition, there can be a distribution again.
Schwartz's core sentence
Each distribution defines a linear mapping , which is the identity
is sufficient and is continuous with regard to the weak - * - topology . That is, if there is a null sequence, then there is also a null sequence in
Conversely, there is exactly one distribution for every linear mapping , so that applies.
This distribution is called the Schwartz core.
literature
Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).