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Solution to the Schauder fixed point problem. (Solution du problème de point fixe de Schauder.) (French) Zbl 0983.54045
Schauder’s conjecture mentioned in the title deals with the question of whether the classical Schauder fixed point theorem (“Let \(C\) be a nonempty convex set in a separated locally convex topological vector space and \(f:C\to C\) a continuous mapping such that \(f(C)\) is contained in a compact subset of \(C\). Then \(f\) has a fixed point.”) really needs the assumption of local convexity. In fact, Schauder himself [Studia Math. 2, 171-180 (1930; JFM 56.0355.01)] believed he had proven the theorem for arbitrary completely metrizable vector spaces but he left an argument for the reader which requires local convexity. Some years later, S. Lefschetz [Topics in topology, Ann. Math. Stud. 10 (1942; Zbl 0061.39303)] proposed another argument which was supposed to work in the non-locally convex case. A closer inspection of the proof, however, reveals that an essential argument will work only in the locally convex case. (As a matter of fact, Lefschetz himself must have felt uneasy about his proof since in the second printing it is stated that a “major correction” has been made in this proof; this reviewer has, however, been unable to detect a version of the first printing of this book and the proof therein.)
The present author now solves this long standing problem by introducing new ideas. If \(X\) is a compact space, denote by \(P(X)\) the set of all probability measures on \(X\) with finite support and by \(P_n(X)\) the set of all \(\mu\in P(X)\) such that the support of \(\mu\) contains at most \(n\) points, so \(P_1(X)\) can be identified with \(X\). Schauder’s theorem then is a consequence of the following more general theorem: If \(X\) is compact then each continuous mapping \(P(X)\to X\) has a fixed point. A result by Shchepin (which can be found as Theorem 3.1.9 in [V.V. Fedorchuk and A. Chigogidze, Absolute retracts and infinite-dimensional manifolds (1992; Zbl 0762.54017)]) reduces the problem to the case where \(X\) is metrizable. A crucial step in the argument then is played by the free topological vector space \(E(X)\) generated by \(X\) and the set \(\mathcal{P}(X)\) of those topologies on \(E(X)\) which make \(E(X)\) into a metric linear space and are coarser than the free topology [the author, Fundam. Math 146, No. 1, 85-99 (1994; Zbl 0817.54014)]. If \(\varphi:Z\to X\) is a map denote by \(\widehat{\varphi}:P(Z)\to P(X)\) the canonical extension. The main step in the proof then rests upon the following result: Let \(X\) be a compact metrizable space. Then there is a countably dimensional compact metrizable space \(Z\) and a continuous mapping \(\varphi:Z\to X\) such that the following holds: If \(\tau\in\mathcal{T}(X)\) and \(\tau'\in\mathcal{T}(Z)\) are such that \(\widehat{\varphi}:(P(Z),\tau')\to(P(X),\tau)\) is continuous then for each \(\tau\)-open cover \(\mathcal{U}\) of \(P(X)\) and each countable locally finite simplicial complex \(N\) and each continuous map \(\xi:N\to X\) there is a continuous map \(\eta:N\to (P(Z),\tau')\) such that \(\widehat{\varphi}\circ\eta\) and \(\xi\) are \(\mathcal{U}\)-close and \(\eta(N)\cup P_2(Z)\) is \(\tau'\)-compact.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
47H10 Fixed-point theorems
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