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antipodal points arbitrary Banach space barycentric coordinates barycentric subdivision Borsuk-Ulam Theorem boundary Cartesian product Cauchy sequence closed ball closed sets closed subset Compact Mappings Theorem compact space compact vector field completely continuous mapping completes the proof condition convex closure COROLLARY corrpact DEFINITION denote equation Extension of Compact f and g field f finite dimensional mapping fixed point property Fixed Point Theorem formula fQ(x free compact field free vector field gQ(x hence Homotopy Extension Theorem inequality intersection Invariance of Domain Lemma Let f Let us put linear mapping mapping F mapping F:X mapping H metric space n-simplexes n-sphere narrow sense norm obtain pair of antipodal positive number proof is completed PROPOSITION prove real numbers retraction satisfied Schauder sequence simplex simplicial mapping singularity free compact singularity free vector solvable subspace Suppose Theorem on Antipodes triangulation uniformly convergent unit ball unit sphere vertex