Solution to the Schauder fixed point problem.
(Solution du problème de point fixe de Schauder.)

*(French)*Zbl 0983.54045Schauder’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.

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.

Reviewer: Christian Fenske (Gießen)