An Infinite-Dimensional Version of the Borsuk–Ulam Theorem

We study the solvability of the equation $a(x)=f(x)$ on a sphere in a Banach space, where $a$ is a closed surjective linear operator and $f$ is an odd $a$-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.