Talk:Schauder fixed-point theorem

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).

Gap in the extension to general Hausdorff topological vector spaces by Cauty
Apparently, in the paper by Cauty (which is quoted as a source of the article), there is a gap. See http://mathoverflow.net/questions/165853/is-schauders-conjecture-resolved. Even the discussion there does not make it completely clear to me whether the general version (without assuming local convexity) is proven or not. I suggest that the article be somewhat rewritten to reflect this. — Preceding unsigned comment added by 134.130.160.208 (talk) 12:46, 20 July 2015 (UTC)

Further updates on http://mathoverflow.net/questions/165853/is-schauders-conjecture-resolved also suggest that Cauty established the proof of Schauder's conjecture in the paper "Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes", published online in 2015. --Saung Tadashi (talk) 17:32, 8 November 2016 (UTC)

Singbal generalization
Someone wrote: B. V. Singbal proved the theorem for the more general case where K may be non-compact; the proof can be found in the appendix of Bonsall's book (see references).

However, the appendix of Bonsall's book contains only this theorem

Theorem. (Singbal).Let E be a locally convex Hausdorff l.t.s., K a non-empty closed convex subset of E, T a continuous mapping of K  into a compact subset of K. Then T has a fixed point in K.

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. --Chyyr (talk) 08:27, 3 December 2020 (UTC)