Smooth and path connected Banach submanifold Σ_r of B(E,F) and a dimension formula in B(ℝⁿ,ℝ^m)

Abstract

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ_r the set of all operators of finite rank r in B(E,F), and Σ^#_r the number of path connected components of Σ_r. It is known that Σ_r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ_r. In this paper,the equality Σ^#_r = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ_r is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T_AΣ_r = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ_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 Σ_r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝⁿ and F = ℝ^m, then Σ_r is a smooth and path connected submanifold of B(ℝⁿ, ℝ^m) and its dimension is dimΣ_r = (m+n)r−r² for each r, 0 <- r < min {n,m}.