Geodesic
Borsuk–Ulam theorem in infinite-dimensional Banach spaces
Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 83-91
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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. 
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B. D. Gel'man. Borsuk–Ulam theorem in infinite-dimensional Banach spaces. Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 83-91. http://geodesic.mathdoc.fr/item/SM_2002_193_1_a2/

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