科研之友
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摘要
Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank It is well-known that B(R-n) is a Banach space as well as an algebra, while B(R-n, R-m) for m not equal n, is a Banach space but not an algebra, meanwhile, It is clear that R(E, F) is neither a Banach space nor an algebra However, in this paper; it is proved that all of them have a common property in geometry and topology, i e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces) Let Sigma(r) be the set of all operators of finite rank r in B(E, F) (or B(R-n, R-m)) In fact, we have that 1) suppose Sigma(r) is an element of B(R-n, R-m), and then Sigma(r) is a smooth and path-connected submanifold of B(R-n, R-m) and dim Sigma(r) = (n + m)r - r(2), for each r is an element of [0, min{n, m}), if m not equal n, the same conclusion for Sigma(r) and its dimension is valid for each r is an element of {0, min{n, m}], 2) suppose Sigma(r) is an element of B(E, F), and dimF = infinity, and then E-r is a smooth and path-connected submanifold of B(E, F) with the tangent space T-A Sigma(r) = {B is an element of B(E, F) : B N(A) subset of R(A)} at each A is an element of Sigma(r) for 0 r <= r < infinity The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(R-n), and more adapted and simple than the elementary transformation method In addition to tensor analysis and application of Thom's famous result for transversility, these will benefit the study of infinite geometry
- 出版日期2010-10
- 单位哈尔滨师范大学; 浙江交通职业技术学院
