Mathematical analysis without gaps
This is a draft of my textbook on mathematical analysis and the areas of mathematics on which it is based. The idea is to fill the gaps in the existing textbooks. Any remarks from readers are welcome.
math.CA↗
arXiv subjects
Sergei Akbarov
Publications and source records attributed to Sergei Akbarov.
Stereotype approximation property for the algebras $C(M)$ of continuous functions on metric spaces
In his previous works the author introduced the notion of the stereotype approximation property as an analog of the classical approximation property transferred to the category ${\tt Ste}$ of stereotype spaces. It is known, that the study of this property is much more difficult than in the classical theory, since the spaces of operators in the category ${\tt Ste}$ are defined in a more complicated way. In this note, it is proved that the algebra $C(M)$ of continuous functions on an arbitrary complete (not necessarily locally compact) metric space $M$ has the stereotype approximation property.
math.FA↗
Envelopes and refinements in categories, with applications to Functional Analysis
An envelope in a category is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone-Čech compactification of a topological space, or universal enveloping algebra of a Lie algebra. Dually, a refinement generalizes operations of "interior enrichment", like bornologification (or saturation) of a locally convex space, or simply connected covering of a Lie group. In this paper we define envelopes and refinements in abstract categories and discuss the conditions under which these constructions exist and are functors. The aim of the exposition is to build a fundament for duality theories of non-commutative groups based on the idea of envelope. The advantage of this approach is that in the arising theories the analogs of group algebras are Hopf algebras. At the same time the classical Fourier and Gelfand transforms are interpreted as envelopes with respect to the prearranged classes of algebras.
math.FA↗