Distributions

Chapter 15Separation of Variables

This chapter is essentially dedicated to show (Theorem 15.10) that the separation of variables is a continuous bijection from ′(Ω₁ Ω₂; E) onto ′(Ω₁; ′(Ω₂; E)) whose inverse mapping, i.e. the regrouping of variables, is sequentially continuous.

This is the diabolical vector-valued kernel theorem of Laurent SCHWARTZ, but for E sequentially complete, which is more general than the quasi-complete he uses, and with a weaker topology on ′(Ω₂; E) which explains why the inverse mapping, that he gets continuous, is only sequentially continuous here.

We prove this by reducing by regularization to the function case and by using the following tools.

1. — The uniform convergence with respect to test functions of the sequences of distributions of distributions form the previous chapter (Theorem 14.6).
2. — The “control” of test functions on by tensor products of test functions (Theorems 15.5 and 15.7).

We finish with the regrouping of variables and their permutation (§ 15.6 and 15.7).

15.1. Tensor products of test functions

This section, like the following two, is ...