Applications

Abstract

In this (and the next) section the main statements of infinite-dimensional distributions theory are stated; this material seems to have a considerable independent interest. Up to the end of the section Z is a Hilbert space, H is a linearly and densely enclosed in Z Hilbert space and the imbedding operator I: H → Z is absolutely summing (= is a Hilbert—Shmidt operator). We denote by T a natural isomorphism of Z on Z* (so that for ∀ z ₁, z ₂ ∈ Z the equalities ${\left( {{z_1},{z_2}} \right)_Z} = {\left( {T{z_1},{z_2}} \right)_H}\mathop {{\text{ }} = }\limits^{def} \left( {T{z_1},{z_2}} \right) = {\left( {T{z_1},{z_2}} \right)_{Z*}}$ hold. T is conjugate with I with respect to the scalar compositions in Z and H).