Revisiting Cauty’s proof of the Schauder conjecture. (English) Zbl 1022.54029

Dobrowolski, Tadeusz

Abstr. Appl. Anal. 2003, No. 7, 407-433 (2003).

The author elaborates on R. Cauty’s proof that every compact convex subset of a metric linear space has the fixed point property [R. Cauty, Fund. Math. 170, No. 3, 231-246 (2001; Zbl 0983.54045)]. In particular, for a compact convex subset \(X\) of a metric linear space, the author gives details about Cauty’s construction of a countable dimensional compactum \(Z\) and a resolution map \(\phi\) from \(Z\) into \(X\). The author also explains how to use the resolution map \(\phi\) to show that compact convex subsets of metric linear spaces have the simplicial approximation property introduced by N. J. Kalton, N. T. Peck and J. W. Roberts in [An \(F\)-Space Sample, London Mathematical Society Lecture Note Series, 89, Cambridge University Press, Cambridge (1984; Zbl 0556.46002)]. The author observes that J. Schauder’s original approach in [Der Fixpunktsatz in Funktionalräumen, Stud. Math. 2, 171-180 (1930)] was actually an attempt to verify the simplicial approximation property, and Kalton et al. [loc. cit.] have shown that the fixed point property follows from the simplicial approximation property.

Reviewer: T.B.Muenzenberger (Manhattan)

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems
55M20	Fixed points and coincidences in algebraic topology
46A16	Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)

Keywords:

Schauder conjecture; metric linear space; fixed point property; simplicial approximation property

Citations:

Zbl 0983.54045; Zbl 0556.46002