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The statement of
Schaefer's theorem is misleading because it gives the impression that T is linear. If that were the case then the set defined in the statement cannot be bounded (multiply x by any scalar).
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Since x is a fixed point of T in K if and only if x is a fixed point in T(K), this theorem still uses the compactness of the set. --
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Apparently, in the paper by Cauty (which is quoted as a source of the article), there is a gap. See
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also suggest that Cauty established the proof of
Schauder's conjecture in the paper "
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http://mathoverflow.net/questions/165853/is-schauders-conjecture-resolved
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http://mathoverflow.net/questions/165853/is-schauders-conjecture-resolved
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Un theoreme de
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However, the appendix of
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B. V. Singbal proved the theorem for the more general case where
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