Schwartz kernel theorem

In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space of test functions. The space itself consists of smooth functions of compact support. Statement of the theorem Let and be open sets in . Every distribution defines a continuous linear map such that for every . Conversely, for every such continuous linear map there exists one and only one distribution such that holds. The distribution is the kernel of the map . Note