SMOOTH AND PATH CONNECTED BANACH SUBMANIFOLD Σ_r OF B(E,F) AND A DIMENSION FORMULA IN B(R~n,R~m)

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear oper- ators from E into F, Σr the set of all operators of finite rank r in B(E,F), and Σ#r the number of path connected components of Σr . It is known that Σr is a smooth Banach submani- fold in B(E,F) with given expression of its tangent space at each A ∈Σr. In this paper,the equality Σ#r = 1 is proved. Consequently, the following theorem is obtained: for any non- negative integer r, Σr is a smooth and path connected Banach submanifol...