To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces

E. V. Manokhin

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 533-541

Résumé

Let's designate through $E(T)$ ($E$ — a Banach space; $T$ — a metric compact set) space of all continuous maps of compact set $T$ in $E$ with sup-norm.Then $E(T)$ — a Banach space. If $E$ is a real axis we there be $E(T)$ to designate through $C(T)$. A. A. Milyutin proved the following theorem.If $K1$ and $K2$ — metric compact sets of continuum cardinality, $E$ — a Banach space, then $E(K1)$ it is isomorphic $E(K2)$.A. A. Milyutin, without knowing about it, in 1951 solved the well-known problem of Banah: whether spaces of continuous functions on a segment and on quadrate are isomorphic. Among works, close to A. A. Milyutin's researches, it is possible to works of M. I. Kadets who has proved topological equivalence of all infinite-dimensional separable Banach spaces works. One of important directions of a functional analysis is geometry of Banach spaces. «The method of equivalent norms» consists in an introduction possibility in a Banach space of the equivalent norm possessing that or other "good" property. The theory of equivalent norms for Banach spaces $C(K)$ continuous functions on metric compact sets is a consequence of the theorem of Milyutin and the theory of separable spaces of Banah. For the case of nonmetrizable compact sets the theory is far from end.The general theory of these compact sets is not present and a little that is known about spaces $C(K)$ for nonmetrizable compact sets with the first countability axiom. Milyutin's theorem has affected researches in this direction. The basic purpose of work is the analysis of influence of the theorem of Milyutin on development of the theory of Banah spaces, especially in one of important directions of a functional analysis — theories of equivalent norms in geometry of Banach spaces. The article presents the results for nonmetrizable compact sets with the first countability axiom, including outcomes received by the author and other mathematicians.

Détail

Keywords: mathematics history, Milyutin's theorem, geometry of spaces of Banaha, the theory of equivalent norms, mathematics of Kharkov.