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SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD Σ_r OF B(E,F) AND A DIMENSION FORMULA IN B(R^n,R^m)
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作者 Jipu Ma 《Analysis in Theory and Applications》 2008年第4期395-400,共6页
Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑r^# the number of path connected components of ∑... Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B(E, F), and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B(E, F) with the tangent space TA∑r = {B E B(E,F) : BN(A) belong to R(A)} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B(R^n,R^m) and its dimension is dim ∑r = (m + n)r- r^2 for each r, 0≤r 〈 min{n,m}. 展开更多
关键词 operator of finite rank smooth Banach submanifolcl path connectivity perturbation analysis of generalized inverse
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