Borsuk–Ulam theorem in infinite-dimensional Banach spaces

The well-known classical Borsuk–Ulam theorem has a broad range of applications to various problems. Its generalization to infinite-dimensional spaces runs across substantial difficulties because its statement is essentially finite-dimensional. A result established in the paper is a natural generalization of the Borsuk–Ulam theorem to infinite-dimensional Banach spaces. Applications of this theorem to various problems are discussed.