March 24, 2006 Fixed Point Theory in Fréchet Space D. P. Dwiggins Systems Support Office of Admissions Department of Mathematical Sciences Analysis Seminar

P : S -> S is a continuous self-mapping Basic Setting X is a topological vector space Also, in most settings, X is complete (If not, X may be considered as a dense subset of its completion.) S is a closed and convex subset of X Either X, S, or F is compact

Fixed Points Finding zeroes of a polynomial A fixed point of such that is any point The search for fixed points is motivated by: Finding the null space for an operator Finding eigenvectors for an operator x 0 is a zero of p(x) iff x 0 is a fixed point of F(x) = x – p(x). Ax 0 = 0 iff x 0 is a fixed point of F(x) = x – Ax. x 0 is an eigenvector for an operator A with corresponding eigenvalue ≠ 0 iff x 0 is a fixed point of F(x) =  Ax, where  = -1.

Topological Vector Space A TVS is a vector (linear) space endowed with a topology, under which the operations of vector addition and scalar multiplication are continuous. The topology might be given by a norm, a quasi-norm, or a separable family of semi-norms. The topology might be defined in terms of measure, and the space might be metrizable. (The chosen topology might also make the TVS locally convex.)

Assumptions on S is always assumed to be: closed under the topology on X convex (a vector property, independent of the topology on X): If X is not assumed to be complete then S must be (which will be true if S is compact). the convex is in S.combination

Assumptions on F If S is not assumed to be compact then F must be completely continuous; i.e. F is both continuous and compact. (This is the usual assumption, with other possible assumptions of the type for some iterate F k – this defines asymptotic FPT.) F is compact if F(A) is compact for every bounded As with the self-mapping condition, this assumption might also be replaced with some alternative, such as requiring F k to be compact for some k > 1. F is continuous and (self-mapping condition)

Finite Dimensional X Brouwer’s Theorem Let S n denote the closed unit ball in Euclidean space R n (note S n is compact). Then any continuous F : S n  S n has a fixed point in S n. L. E. J. Brouwer, Math. Annalen 71 (1911) There are many ways to prove this result, including a purely combinatoric argument using mappings on finite- dimensional simplices. Moreover, this theorem also holds if S n is replaced by any finite-dimensional H n which is homeomorphic to the closed unit ball.

Infinite Dimensional X Schauder-Tychonov Theory To extend from finite to infinite dimensional space, what needs to be determined are the types of space X for which every continuous self-mapping F : S  S on any closed convex compact subset S  X has a fixed point in S. Such spaces X are called fixed point spaces, and Banach spaces (complete normed linear spaces) are all fixed point spaces. However, the earliest results were set in spaces which were more general, and which include Banach space as one specific example.

Infinite Dimensional X Schauder-Tychonov Theory Schauder’s Theorem (Studia Mathemtaica v.2, 1930): Any complete quasi-normed space is a fixed point space. Tychonov’s Theorem (Math. Annalen v.111, 1935): Any complete locally convex TVS is a fixed point space. Most authors who cite this theorem assume X to be a Banach space, which Schauder did not do, and which he mentioned in a footnote, that the metric he was using (a quasi-norm) does not possess the homogeneity of a norm, and thus he was not working in a “B-space”. Since our basic setting assumes a complete TVS, this theorem might be viewed as “best possible”. However, there are quasi-normed spaces which are not locally convex, and so Schauder’s Theorem remains independent.

Quasi-normed Space versus LCTVS Let X be a complete metric linear space, for which the linear operations are continuous with respect to the metric (i.e. X is a TVS). Then X becomes a quasi-normed space if the metric is translation invariant: X would become a Banach space if the metric were also homogeneous:

Quasi-normed Space versus LCTVS A LCTVS is a TVS whose topology can be generated from a separable family of seminorms (Yosida F. A. pp 23-26). If a LCTVS is metrizable, then its topology can be obtained from a countable family of seminorms {  n }, from which a quasi-norm is obtained. However, there are examples of non-metrizable LCTVS (Yosdia pg 28).

Quasi-normed Space versus LCTVS Thus, Tychonov’s theorem holds in spaces for which Schauder’s theorem does not hold (any non-metrizable complete LCTVS). Yosida also gives an example of a complete quasi-normed space which is not locally convex, meaning there are spaces in which Schauder’s theorem holds but not Tychonov’s theorem. The components of this example appear on pages 38, 117, and 108 (in that order) of Yosida’s text.

Quasi-normed Space versus LCTVS Let Q denote the class of all measurable functions x : [0,1]  C which are defined a.e. on [0,1] (C is the set of complex numbers). Define a quasi-norm on Q by Then Q is complete (Yosida pg 38) but is not locally convex. To prove this, it is first argued that the dual of Q (denoted by Q) consists only of the zero functional (pg 117). This also gives a translation invariant metric  by defining

Quasi-normed Space versus LCTVS Next, consider the subspace M consisting of all x  Q such that x(0) = 0. Now let y  Q be given with y(0)  0 (and so y  M). Then, as a consequence of the Hahn-Banach theorem (found in Yosida’s text on pg 108), if Q were locally convex there would be a continuous linear function f  Q such that f(y) > 1. This contradicts Q consisting only of the zero functional, and so Q cannot be locally convex.

What is a Fréchet Space? Wikipedia: In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special TVS’s. They are generalizations of Banach spaces, which are complete with respect to the metric induced by the norm. Fréchet spaces, in contrast, are locally convex spaces which are complete with respect to a translation invariant metric, which may be generated by a countable family of semi-norms. Every Banach space is a Fréchet space, which in general has a more complicated topological structure due to lack of a norm, but in which important results such as the open mapping theorem and the Banach- Steinhaus theorem still hold. Other examples of Fréchet spaces include infinitely differentiable functions on compact sets (the seminorms use bounds on the k th derivative over the compact set) and the space consisting of sequences of real numbers, with the k th seminorm being the absolute value of the k th term in the sequence.

What is a Fréchet Space? Yosida (F. A. page 52) defines a Fréchet space to be a complete quasi- normed space (this type of space was used in the proof of Schauder’s theorem ), but notes that “Bourbaki” defines a Fréchet space as a complete LCTVS which is metrizable. Every metrizable LCTVS defines a quasi- norm, but not every quasi-normed space is locally convex. Grothendieck (TVS 1973, page 177) states there are some metrizable LCTVS’s which are not quasi-normable, but uses a quite different definition of what is a quasi-norm. It also appears some authors have used the term Fréchet space to denote a complete LCTVS, metrizable or not. This is the type of space used in Tychonov’s fixed point theorem. Finally, some authors have used the term Fréchet space to denote a space whose topology may be defined in terms of sequences, without any reference to a metric or even a vector space. See Franklin, “Spaces in which Sequences Suffice,” Fund. Math. 57 (1965).

M What is a Fréchet Space? Schauder space = complete quasi-normed space, locally convex or not Tychonov space = complete LCTVS, metrizable or not Wikipedia: Dwiggins: ST B = Banach Space M = complete metric space as a TVS, whether or not the metric is translation invariant B

Asymptotic Fixed Point Theory One way to remove the self-mapping condition F : S  S is instead to require F k : S  S, where F 2 (x) = F(F(x)), F 3 (x) = F(F 2 (x)), et cetera, for some k > 1. Any fixed point theorem using iterates of the mapping is said to be of asymptotic type. Unfortunately it is often just as difficult to require F k : S  S for some k > 1 as it is for k = 1. Instead, consider a sequence of sets S 0  S 1  S 2, where eventual iterates of F map S 1 into S 0, and all iterates of F map S 1 into S 2. In this setting, F is not a self-mapping on S 0, but eventually every point in S 0 ends up back in S 0 as it travels along an orbit of F, and even early in the orbit the point is never “too far away” from where it started.

Asymptotic FPT Horn’s Theorem Let S 0  S 1  S 2 be convex subsets of a Fréchet space X, with S 0 and S 2 both compact and S 1 open relative to S 2. Then F has a fixed point in S 0. Let F : S 2  X be a continuous map such that, for some m > 1, S0S0 S1S1 S2S2 W. A. Horn, Trans. AMS 149 (1970). Note: Horn assumed X to be a Banach space in his paper. However, if one lemma is re-written and the symbol for the norm is everywhere replaced with a metric then his proof still holds.