Compact - de ned decompositions of spaces with applications to calculus of

Last decade, at rst the variational problems in Sobolev spaces, later the Radon-Nikodym problem in vector integration, and some other problems leads us to the new function characteristics, both scalar and convex compact, which are closely connected to a certain decomposition of the space under consideration. That decomposition is generated by the Banach subspaces spanned by the all absolutely convex compacta from the initial space. The structure of such decompositions of Frechet spaces is like to one of the 3⁄4di racting screen¿ in physics. The spectrum of such subspaces doesn't change the linear e ects, but shows evidently enough some new nonlinear e ects. It seems that the researches with the help of compact-de ned decompositions can be used for many actual problems of the modern analysis. At least, my own program is extensive enough (may be, too much). Some works about reference. The papers 1 5 are devoted mainly to variational problems. That researches were realized jointly with E.Bozhonok. The papers 6 9 are devoted mainly to vector integration. That researches were realized jointly with F.Stonyakin. Finally, the paper 10 contains compact analog of Banach-Zaretsky theorem.