Open AccessOn Complexification of Real Spaces and Its Manifestations in the Theory of Bochner and Pettis Integrals
Author(s) -
M. Elena Luna-Elizarrarás,
F Ram ́ırez-Reyes,
Ф Рамирез-Рейес,
Michael Shapiro,
Michael Shapiro
Publication year - 2018
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2018-64-4-706-722
Subject(s) - mathematics , hilbert space , pure mathematics , multiplication (music) , banach space , norm (philosophy) , bochner space , set (abstract data type) , space (punctuation) , linear operators , topological tensor product , work (physics) , banach manifold , mathematical analysis , lp space , functional analysis , computer science , combinatorics , law , chemistry , operating system , biochemistry , political science , bounded function , programming language , gene , engineering , mechanical engineering
This work is a continuation of our work [12] where we considered linear spaces in the following two situations: a real space admits a multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with a multiplication by complex scalars. We studied there also how they manifest themselves when the initial space possesses additional structures: topology, norm, inner product, as well as what happens with linear operators acting between such spaces. Changing the linearities of the linear spaces unmasks some very subtle properties which are not so obvious when the set of scalars is not changed. In the present work, we follow the same idea considering now Bochner and Pettis integrals for functions ranged in real and complex Banach and Hilbert spaces. Finally, this leads to the study of strong and weak random elements with values in real and complex Banach and Hilbert spaces, in particular, some properties of their expectations.