
To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces
E. V. Manokhin^{} ^{} Financial University under the Government of the Russian Federation, Tula branch
Abstract:
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 supnorm. 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 wellknown 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 infinitedimensional 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.
Keywords:
mathematics history, Milyutin's theorem, geometry of spaces of Banaha, the theory of equivalent norms, mathematics of Kharkov.
DOI:
https://doi.org/10.22405/222683832018192533541
Full text:
PDF file (319 kB)
References:
PDF file
HTML file
UDC:
51(091) Received: 10.06.2018 Accepted:17.08.2018
Citation:
E. V. Manokhin, “To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces”, Chebyshevskii Sb., 19:2 (2018), 533–541
Citation in format AMSBIB
\Bibitem{Man18}
\by E.~V.~Manokhin
\paper To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 2
\pages 533541
\mathnet{http://mi.mathnet.ru/cheb672}
\crossref{https://doi.org/10.22405/222683832018192533541}
\elib{http://elibrary.ru/item.asp?id=37112172}
Linking options:
http://mi.mathnet.ru/eng/cheb672 http://mi.mathnet.ru/eng/cheb/v19/i2/p533
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  60  Full text:  11  References:  3 
