Spaces not containing ℓ 1 have weak aproximate fixed point property
aa r X i v : . [ m a t h . F A ] J u l SPACES NOT CONTAINING ℓ HAVE WEAK APROXIMATE FIXEDPOINT PROPERTY
OND ˇREJ F.K. KALENDA
Abstract.
A nonempty closed convex bounded subset C of a Banach space is said tohave the weak approximate fixed point property if for every continuous map f : C → C there is a sequence { x n } in C such that x n − f ( x n ) converge weakly to 0. We prove inparticular that C has this property whenever it contains no sequence equivalent to thestandard basis of ℓ . As a byproduct we obtain a characterization of Banach spaces notcontaining ℓ in terms of the weak topology. Introduction and main results
Let X be a real Banach space and C a nonempty closed convex bounded subset of X . The set C is said to have the approximate fixed point property (shortly afp property )if for every continuous mapping f : C → C there is a sequence { x n } in C such that x n − f ( x n ) →
0. The set C is said to have the weak approximate fixed point property (shortly weak afp property ) if for every continuous mapping f : C → C there is a sequence { x n } in C such that the sequence { x n − f ( x n ) } weakly converges to 0.The study of these notions was started by C. S. Barroso [1] in topological vector spaceswhere, in particular, the weak afpp for weakly compact convex subsets of Banach spaceswas proved, and after by C.S. Barroso and P.-K. Lin [2] in Banach spaces for generalbounded, closed convex sets with emphazis on geometrical aspects.Our terminology follows [2]. Anyway, it is worth to remark that the notion of the afpproperty in this context does not have a good meaning. Indeed, if C is compact, thenany continuous selfmap of C has even a fixed point by Schauder’s theorem (see e.g. [4,p. 151, Theorem 183]). If C is not compact, then it does not have the afp property by aresult of P.-K. Lin and Y. Sternfeld [5, Theorem 1]. Anyway it may have a sense in caseof non-complete X or non-closed C . A Lipschitz version of this property is studied in [5].For the weak afp property the situation is different:A Banach space X is said to have the weak approximate fixed point property if eachnonempty closed convex bounded subset of X has the weak afp property.This notion was studied by C.S. Barroso and P.-K. Lin in [2]. They proved that Asplundspace do have the weak afp and asked in Problem 1.1 whether the same is true for spacesnot containing ℓ . In the present paper we answer this question affirmatively. This is thecontent of the following theorem. Theorem 1.1.
Let X be a Banach space. Then X has the weak approximate fixed pointproperty if and only if X contains no isomorphic copy of ℓ . Mathematics Subject Classification.
Key words and phrases. weak approximate fixed point property; ℓ sequence; Fr´echet-Urysohn space.The research was supported in part by the grant GAAV IAA 100190901 and in part by the ResearchProject MSM 0021620839 from the Czech Ministry of Education. This theorem is an immediate consequence of the following more general theorem:
Theorem 1.2.
Let X be a Banach space and C a nonempty closed convex bounded subsetof X . Then the following assertions are equivalent. (1) Each nonempty closed convex subset of C has the weak approximate fixed pointproperty. (2) C contains no sequence equivalent to the standard basis of ℓ . Let us recall that a bounded sequence { x n } is equivalent to the standard basis of ℓ ifthere is a constant c > N ∈ N and any choice of a , . . . , a N ∈ R wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X n =1 a n x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ c N X n =1 | a n | . It means that the mapping T : ℓ → X defined by T ( { a n } ) = P ∞ n =1 a n x n is an isomorphicembedding. Such sequences { x n } will be called ℓ -sequences .The implication (1) ⇒ (2) is known to be true. Indeed, suppose that { x n } is an ℓ -sequence contained in C . Set D to be the closed convex hull of the set { x n : n ∈ N } . Let T be the mapping defined in the previous paragraph. Then Y = T ( ℓ ) is a subspace of X which is isomorphic to ℓ and contains D . So, by Schur’s theorem (see e.g. [4, p. 74,Theorem 99]), weakly convergent sequences in Y are norm convergent. So, if D had theweak afp property, it would have the afp property as well. But it is impossible by thealready quoted [5, Theorem 1] as D is not compact.We remark that Theorem 1.2 immediately implies that weakly compact sets have weakafp property which also follows from a result of C.S. Barroso [1, Theorem 3.1].We finish this section by recalling and commenting two results from [2]. Lemma 1.3.
Let X be any Banach space, C any nonempty closed convex bounded subsetof X and f : C → C any continuous mapping. Then the point is in the weak closure ofthe set { x − f ( x ) : x ∈ C } . This lemma is proved in [2, Lemma 2.1] using Brouwer’s fixed point theorem andparacompactness of metric spaces.
Lemma 1.4.
Let X be any Banach space, C any nonempty closed convex bounded subsetof X and f : C → C any continuous mapping. Then there is a nonempty closed convexseparable subset D ⊂ C with f ( D ) ⊂ D . This is easy and is proved in the second part of the proof of Theorem 2.2 in [2].In view of Lemma 1.3 to prove the weak afp property one needs to reach the point 0 bya limit of a sequence, not just by a limit of a net. In [2, Theorem 2.2] it is done by usingimplicitly the metrizability of the weak topology on bounded sets of a separable Asplundspace. We show that it is also possible under the weaker assumption that the space doesnot contain a copy of ℓ . Topological results which enable us to do so are contained inthe following section.2. ℓ -sequences and Fr´echet-Urysohn property of the weak topology Let us recall that a topological space T is called Fr´echet-Urysohn if the closures ofsubsets of T are described using sequences, i.e. if whenever A ⊂ T and x ∈ T is such that x ∈ A , there is a sequence { x n } in A with x n → x . Metrizable spaces are Fr´echet-Urysohn EAK APROXIMATE FIXED POINT PROPERTY 3 but there are many nonmetrizable Fr´echet-Urysohn spaces (for examples see the resultsbelow).We will need the following deep result of J. Bourgain, D.H. Fremlin and M. Talagrand[3, Theorem 3F]:
Theorem 2.1.
Let P be a Polish space (i.e., a separable completely metrizable space).Denote by B ( P ) the space of all real-valued functions on P which are of the first Baireclass and equip this space with the topology of pointwise convergence. Suppose that A ⊂ B ( P ) is relatively countably compact in B ( P ) (i.e., each sequence in A has a clusterpoint in B ( P ) ). Then the closure A of A in B ( P ) is compact and Fr´echet-Urysohn. In fact, we need a slightly weaker version formulated in the following corollary.
Corollary 2.2.
Let P be a Polish space and A be a set of real-valued continuous functionson P . Suppose that each sequence in A has a pointwise convergent subsequence. Then theclosure of A in R P is a Fr´echet-Urysohn compact space contained in B ( P ) .Proof. A is obviously contained in B ( P ). Moreover, let ( f n ) be any sequence in A . Bythe assumption there is a subsequence ( f n k ) pointwise converging to some function f . Asthe functions f n k are continuous, the limit function f is of the first Baire class. Hence, itis a cluster point of ( f n ) in B ( P ). So, A is relatively countably compact in B ( P ). Theassertion now follows from Theorem 2.1. (cid:3) Now we are ready to prove the following proposition which can be viewed as an im-provent of a result due to E. Odell and H.P. Rosenthal [6] on characterization of separablespaces not containing ℓ . We note that we use the results of [3] and this paper waspublished three years after [6]. Proposition 2.3.
Let X be a Banach space and C be a bounded subset of X . If C isnorm-separable and contains no ℓ -sequence, then the set κ ( C − C ) w ∗ = { κ ( x − y ) : x, y ∈ C } w ∗ is Fr´echet-Urysohn when equipped with the weak* topology, where κ denotes the canonicalembedding of X into X ∗∗ . In particular, C − C w = { x − y : x, y ∈ C } w is Fr´echet-Urysohn when equipped with the weak topology.Proof. As the closed linear span of C is separable, we can without loss of generalitysuppose that X is separable. Further we have:Each sequence in C − C has a weakly Cauchy subsequence. ( ∗ )Indeed, let { z n } be a sequence in C − C . Then there are sequences { x n } and { y n } in C suchthat z n = x n − y n for each n ∈ N . As C contains no ℓ -sequence, by Rosenthal’s theorem[7] there is a subsequence { x n k } of { x n } which is weakly Cauchy. Using Rosenthal’stheorem once more, we get a subsequence { y n kl } of y n k which is weakly Cauchy. Then { z n kl } = { x n kl − y n kl } is a weakly Cauchy subsequence of { z n } . This completes the proofof (*).Further, denote by K the dual unit ball ( B X ∗ , w ∗ ) equipped with the weak* topology.Then K is a metrizable compact space. Denote by r the mapping r : X ∗∗ → R K definedby r ( F ) = F | K for F ∈ X ∗∗ . Then we have: O.F.K. KALENDA (i) r is a homeomorphism of ( X ∗∗ , w ∗ ) onto r ( X ∗∗ ).(ii) r ◦ κ is a homeomorphism of ( X, w ) onto r ( κ ( X )).(iii) The functions from r ( κ ( X )) are continuous on K .Set M = r ( κ ( C − C )). Then M is a uniformly bounded sets of continuous functions on K . Moreover, by (*) any sequence from M has a pointwise convergent subsequence. ByCorollary 2.2 the closure of M in R K is a Fr´echet-Urysohn compact subset of B ( K ). Butthis closure is equal to r (cid:16) κ ( C − C ) w ∗ (cid:17) . It follows that κ ( C − C ) w ∗ is Fr´echet-Urysohnin the weak* topology. This completes the proof of the first statement.Further, to show the ‘in particular case’ it is enough to observe that the set κ ( C − C ) w ∗ contains κ (cid:0) C − C w (cid:1) , hence C − C w is Fr´echet-Urysohn in the weak topology. (cid:3) As a corollary we get the following characterization of spaces not containing ℓ : Theorem 2.4.
Let X be a Banach space. Then the following assertions are equivalent. (1) X contains no isomorphic copy of ℓ . (2) Each bounded separable subset of X is Fr´echet-Urysohn in the weak topology. (3) For each separable subset A ⊂ X there are relatively weakly closed subsets A n , n ∈ N , of A such that A = S n ∈ N A n and each A n is Fr´echet-Urysohn in the weaktopology. Note that the assertion (3) is a topological property of the space (
X, w ) (as normseparability coincides with weak separability).
Proof.
The implication (1) ⇒ (2) follows from Proposition 2.3.The implication (2) ⇒ (1) follows from the fact that the unit ball of ℓ is not Fr´echet-Urysohn (as 0 is in the weak closure of the sphere and the sphere is weakly sequentiallyclosed by the Schur theorem).The implication (2) ⇒ (3) is trivial if we use the fact that a closed ball is weakly closed.Let us prove (3) ⇒ (2). To show (2) it is enough to prove that the unit ball of anyclosed separable subspace of X is Fr´echet-Urysohn in the weak topology. Let Y be such asubspace. Let Y n , n ∈ N , be the cover of Y provided by (3). As each Y n is weakly closed,it is also norm-closed. By Baire category theorem some Y n has a nonempty interior in Y ,so it contains a ball. We get that some ball in Y is Fr´echet-Urysohn, so the unit ball hasthis property as well. (cid:3) It is worth to compare the previous theorem with a similar characterization of Asplundspaces:
Theorem 2.5.
Let X be a Banach space. Then the following assertions are equivalent. (1) X is Asplund. (2) Each bounded separable subset of X is metrizable in the weak topology. (3) For each separable subset A ⊂ X there are relatively weakly closed subsets A n , n ∈ N , of A such that A = S n ∈ N A n and each A n is metrizable in the weaktopology. We recall that X is Asplund if and only if Y ∗ is separable for each separable subspace Y ⊂ X . The equivalence of (1) and (2) follows from the well-known fact that the unit ballof Y is metrizable in the weak topology if and only if Y ∗ is separable. The equivalence of(2) and (3) can be proved similarly as corresponding equivalence in the previous theorem. EAK APROXIMATE FIXED POINT PROPERTY 5
Remark 2.6.
There is no analogue of Theorem 2.4 for convex sets. Indeed, let X = ℓ and C be the closed convex hull of the standard basis. Then C contains an ℓ -sequencebut is Fr´echet Urysohn in the weak topology. In fact, it is even metrizable as it is easy tosee that on the positive cone of ℓ the weak and norm topologies coincide.3. Proof of Theorem 1.2
It remains to prove the implication (2) ⇒ (1). Let X be a Banach space, C ⊂ X anonempty closed convex bounded set containing no ℓ -sequence and f : C → C be acontinuous mapping. Let D be the set provided by Lemma 1.4. Then D is separable andcontains no ℓ -sequence. By Lemma 1.3 the point 0 is in the weak closure of { x − f ( x ) : x ∈ D } . By Proposition 2.3 there is a sequence from this set weakly converging to 0.This completes the proof. (cid:3) Remark 3.1.
We stress the difference between approximation in the norm and in theweak topology. Suppose that X is a Banach space, C ⊂ X a nonempty closed convexbounded set and f : C → C a continuous mapping.For approximation in the norm, we have the equivalence of the following three condi-tions: • There is a sequence { x n } in C such that x n − f ( x n ) → • The point 0 is in the norm-closure of the set { x − f ( x ) : x ∈ C } . • inf {k x − f ( x ) k : x ∈ C } = 0.These three statements are trivially equivalent (by properties of metric spaces) and arerather strong. For the weak topology the situation the situation is different. First, thereis no analogue of the third condition. Secondly, the analogue of the second one is satisfiedallways by Lemma 1.3. But the analogue of the first one is not satisfied allways, as theweak topology is not in general described by sequences. Acknowledgement
The author is grateful to Barry Turett for informing him about the problem. Thanksare also due to Mari´an Fabian for providing the contact to Barry Turett.
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Department of Mathematical Analysis, Faculty of Mathematics and Physic, CharlesUniversity, Sokolovsk´a 83, 186 75, Praha 8, Czech Republic
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