Использование гомологических методов на базе итерированных спектров в функциональном анализе

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Frechet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quotient spaces. Moreover, for H-space an unproved variant of the closed graph theorem holds true. Homological methods are used for proving of theorems of vanishing at zero for first derivative of Hausdorff limit functor: Haus1(X)=0.

Inductive and projective limits of smooth topological vector spaces

In J. Math. Mech. 15 (1966), 877–898, Bonic and Frampton have laid the foundation for a general theory of smoothness of Banach spaces. In this paper, we shall study one aspect of the smoothness of topological vector spaces, namely, the relationship between smoothness and inductive and protective limits of topological vector spaces. As a consequence, we obtain smoothness results for nuclear spaces and some Montei spaces.

Inductive and projective limits of normed spaces

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.

Lifting Inductive and Projective Limits

In this paper, which is the fourth in a series of articles (11, 12, 13) on universal solutions in categories, a relationship between inductive limits and final structures (or projective limits and initial structures) is studied. The problems to be encountered are illustrated by the following example.