In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with...In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.展开更多
The 21st century is the era of human recreation. What leisure and sport bring to people is bound to be a civilized, healthy, happy, and new life patterns. Throughout the development of sports and leisure history, we c...The 21st century is the era of human recreation. What leisure and sport bring to people is bound to be a civilized, healthy, happy, and new life patterns. Throughout the development of sports and leisure history, we can see that in the era of leisure sports agricultural, civilization reflects a more casual game. Strictly speaking, leisure sport is a product of the birth of modern industrial civilization. But today, it has become a recreational sport entirely necessities of life, recreation is the ultimate ownership of human beings, but also one of the best state of human existence, and therefore we have reason to believe that in the near future, leisure Sports will bring a self-liberation and ultimate human freedom.展开更多
基金supported by the Natural Science Foundation of China (12271402)the Natural Science Foundation of Tianjin City (22JCYBJC00420)。
文摘In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.
文摘The 21st century is the era of human recreation. What leisure and sport bring to people is bound to be a civilized, healthy, happy, and new life patterns. Throughout the development of sports and leisure history, we can see that in the era of leisure sports agricultural, civilization reflects a more casual game. Strictly speaking, leisure sport is a product of the birth of modern industrial civilization. But today, it has become a recreational sport entirely necessities of life, recreation is the ultimate ownership of human beings, but also one of the best state of human existence, and therefore we have reason to believe that in the near future, leisure Sports will bring a self-liberation and ultimate human freedom.
