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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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▲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 ==
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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 "[https://www.degruyter.com/abstract/j/crll.ahead-of-print/crelle-2014-0134/crelle-2014-0134.xml Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes]", published online in 2015. --[[User:Saung Tadashi|Saung Tadashi]] ([[User talk:Saung Tadashi|talk]]) 17:32, 8 November 2016 (UTC)
:the link to Cauty's article is dead. [[User:Logicdavid|Logicdavid]] ([[User talk:Logicdavid|talk]]) 16:46, 16 April 2025 (UTC)
:Moreover, the article is not referenced at all in the wikipedia article, is it? [[User:Logicdavid|Logicdavid]] ([[User talk:Logicdavid|talk]]) 16:50, 16 April 2025 (UTC)
==Singbal generalization ==
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<blockquote>
'''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.
</blockquote>
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.
--[[User:Chyyr|Chyyr]] ([[User talk:Chyyr|talk]]) 08:27, 3 December 2020 (UTC)
== References missing in text ==
The article states
<blockquote>
[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.
</blockquote>
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? ''[[User:Logicdavid|Logicdavid]] ([[User talk:Logicdavid|talk]]) 16:36, 16 April 2025 (UTC)''
== 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. [[User:Logicdavid|Logicdavid]] ([[User talk:Logicdavid|talk]]) 16:39, 16 April 2025 (UTC)
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