A generalization of antipodal point theorems for set-valued mappings

Let U be a bounded symmetric open neighborhood of the origin of Rm+k (k >= 1). We shall prove a generalization of the Borsuk's antipodal theorem for an admissible mapping phi : partial derivative(U) over bar -> R-m and the related topic. We shall generalize the theorem for the case of a bounded symmetric open neighborhood U of the origin of an infinite dimensional normed space E. The Borsuk-Ulam theorem shall be studied for the case of a bounded symmetric open neighborhood U of the origin of an infinite dimensional normed space E.