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)[reply]

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)[reply]

the link to Cauty's article is dead. Logicdavid (talk) 16:46, 16 April 2025 (UTC)[reply]

Moreover, the article is not referenced at all in the wikipedia article, is it? Logicdavid (talk) 16:50, 16 April 2025 (UTC)[reply]

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)[reply]

References missing in text

The article states

[Schauder's Fixed Point Theorem] asserts that if C is a nonempty convex closed subset of a Hausdorff topological vector space and is f continuous mapping of C into itself such that is contained in a compact subset of C, then f has a fixed point.

That's not what is called Schauder's Fixed Point Theorem in most texts in the literature, namely because the latter usually concern Banach spaces. The theorem as stated here is moreover without citation. This is a significant problem, in my view. Where is this purported theorem from, exactly? Logicdavid (talk) 16:36, 16 April 2025 (UTC)[reply]

A ghost Theorem

The article also coins the expression Leray–Schauder theorem without telling us which theorem this is, where it can be found, what it states precisely. Logicdavid (talk) 16:39, 16 April 2025 (UTC)[reply]