Geodesic
An Infinite-Dimensional Version of the Borsuk--Ulam Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 1-5.

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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.
Keywords: closed surjective operator, compact map, operator equation. 
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B. D. Gel'man. An Infinite-Dimensional Version of the Borsuk--Ulam Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 4, pp. 1-5. http://geodesic.mathdoc.fr/item/FAA_2004_38_4_a0/

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