Analogs of weyl inequalities and the trace theorem in banach space

Let A be a completely continuous operator acting on the Banach space, let be the complete system of its eigenvalues (with regard for multiplicity) and let be the distance from A to the set of all operators of range dimension not greater than n. If then, where sp(A) is a functional which is linear on the set of operators satisfying condition (1) (and continuous in a certain topology) and which coincides with its trace for finite-dimensional A. The proof of this theorem is based on certain analogs of the famous Weyl inequalities.