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On isometric extension problem between two unit spheres

Science China Mathematics 2009年第10期52卷 2069-2083页

作者： Ding GuangGui School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China School of Mathematical Sciences and LPMC Nankai University Tianjin China

In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.

关键词： normed space isometric extension isometric mapping 1-Lipschitz mapping 46A22 46b02 46b04 46b20 46b22

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Several applications of the theory of random conjugate spaces to measurability problems

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Science China Mathematics 2007年第5期50卷 737-747页

作者： Tie-xin GUO Department of Mathematics, School of Science, beijing University of Aeronautics and Astronautics, beijing 100083, China Department of Mathematics School of Science Beijing University of Aeronautics and Astronautics Beijing China

The central purpose of this paper is to illustrate that combining the recently developed theory of random conjugate spaces and the deep theory of banach spaces can, indeed, solve some difficult measurability problems which occur in the recent study of the Lebesgue (or more general, Orlicz)-bochner function spaces as well as in a slightly different way in the study of the random functional analysis but for which the measurable selection theorems currently available are not applicable. It is important that this paper provides a new method of studying a large class of the measurability problems, namely first converting the measurability problems to the abstract existence problems in the random metric theory and then combining the random metric theory and the relative theory of classical spaces so that the measurability problems can be eventually solved. The new method is based on the deep development of the random metric theory as well as on the subtle combination of the random metric theory with classical space theory.

关键词： random normed module random conjugate space measurability problem 46b09 46H25 46A22 46b22 60H25

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