The fixed point property under renorming in some classes of Banach spaces

Nonlinear Analysis 72 (2010) 1409–1416

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The fixed point property under renorming in some classes of Banach spaces T. Domínguez Benavides ∗ , S. Phothi Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080-Sevilla, Spain

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Article history: Received 20 April 2009 Accepted 5 August 2009 MSC: primary 46B20 47H09 secondary 47H10 Keywords: Nonexpansive mapping Fixed point Banach space Equivalent norm Baire category Residual set

abstract Assume that Y is a Banach space such that R(Y ) < 2 where R(·) is García-Falset’s coefficient, and X is a Banach space which can be continuously embedded in Y . We prove that X can be renormed to satisfy the weak Fixed Point Property (w-FPP). On the other hand, assume that K is a scattered compact topological space such that K (ω) = ∅ and C (K ) is the space of all real continuous functions defined on K with the supremum norm. We will show that C (K ) can be renormed to satisfy R(C (K )) < 2. Thus, both results together imply that any Banach space which can be continuously embedded in C (K ), K as above, can be renormed to satisfy the w-FPP. These results extend a previous one about the w-FPP under renorming for Banach spaces which can be continuously embedded in c0 (Γ ). Furthermore, we consider a metric in the space P of all norms in C (K ) which are equivalent to the supremum norm and we show that for almost all norms in P (in the sense of porosity) C (K ) satisfies the w-FPP. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Assume that (X , k · k) is a Banach space. The most common aim of the Renorming Theory is to find an equivalent norm | · | which satisfies (or which does not satisfy) certain specific properties. A detailed account of this theory can be found in the monographs [1–3]. This paper focuses on the Renorming Theory in connection with the Fixed Point Theory. We recall that a mapping T defined from a metric space M into M is said to be non-expansive if d(Tx, Ty) ≤ d(x, y) for every x, y ∈ M. It is well known that the Contractive Mapping Principle fails for non-expansive mappings, but many results about existence of fixed point for non-expansive mappings in weakly compact convex sets of some classes of Banach spaces have been obtained (see, for instance, [4] or [5]). It is usually said that a Banach space X satisfies the weak Fixed Point Property (w-FPP) if for every convex weakly compact subset C of X , each non-expansive mapping T : C → C has a fixed point. Many geometrical properties of X (uniform convexity, uniform smoothness, uniform convexity in every direction, uniform non-squareness, normal structure, etc) are known to imply the w-FPP, but no characterization of the w-FPP in terms of these properties is known. Therefore, we can regard the w-FPP as an intrinsic property of a Banach space. It is relevant to note that the wFPP is not preserved under isomorphisms. Indeed, it is well known that the space L1 ([0, 1]) does not satisfy the w-FPP as proved by Alspach [6]. However this space (and any separable Banach space) can be renormed to have normal structure [7] and so the w-FPP [8]. Thus, a very natural question in Renorming Theory and Fixed Point Theory would be the following: let X be a Banach space. Is it possible to renorm X so that the resultant space has the w-FPP? This is not generally the case. Indeed, Partington [9,10] has proved that every renorming of `∞ (Γ ) for Γ uncountable and any renorming of `∞ /c0 contain an isometric copy of `∞ and so they fail the w-FPP (again due to Alspach example). Thus, it would be interesting to identify some classes of Banach spaces which can be renormed to satisfy the w-FPP. For instance the following question

∗

Corresponding author. E-mail addresses: [email protected] (T.D. Benavides), [email protected] (S. Phothi).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.08.024

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T.D. Benavides, S. Phothi / Nonlinear Analysis 72 (2010) 1409–1416

which appears in [11] (Open Question VI) and [1] (problem VII.3) remained unanswered for a long time: can any reflexive Banach space be renormed to satisfy the w-FPP? In [12] it is shown that this is indeed the case. Actually, the following result is proved in [12]: assume that X is a Banach space such that there exists a bounded one-to-one linear operator from X into c0 (Γ ). Then, X has an equivalent norm which satisfies the w-FPP. The proof of the result is strongly based upon some specific properties of the space c0 (Γ ), specially the equality R(c0 (Γ )) = 1, where R(·) is García-Falset’s coefficient [13]. It must be noted that any Banach space Y such that R(Y ) < 2 satisfies the w-FPP. Thus, it would be natural to try and extend the above result to any Banach space which can be embedded in more general Banach spaces than c0 (Γ ), but still satisfying R(Y ) < 2. In Section 2 of this paper, we actually prove this extension in the following sense: assume that Y is a Banach space such that R(Y ) < 2 and X is a Banach space which can be continuously embedded in Y . Then, X can be renormed to satisfy the w-FPP. On the other hand, if we endow Γ with the discrete topology and denote by K the one-point compactification of Γ , then c0 (Γ ) is isometrically contained in C (K ), and K is a topological compact space which satisfies K (2) = ∅. Thus, any space which can be continuously embedded in c0 (Γ ) can also be embedded in C (K ) where K is a scattered compact topological space such that K (ω) = ∅. Since C (K ) satisfies the w-FPP [14] when K is a scattered compact topological space K such that K (ω) = ∅, another natural question would be the following: assume that X is a Banach space which can be continuously embedded in C (K ) for some K as above. Can X be renormed to satisfy the w-FPP? In Section 3 we prove a result that shows this is indeed the case. Nominally we prove the following: let C (K ) be the space of real continuous functions defined on a scattered compact topological space K such that K (ω) = ∅. Then, it can be renormed in such a way that R(C (K ), k · k) < 2 where k · k is the new norm. A better understanding of the relevance of this result is achieved noting that in the metrizable case, if K (ω) = ∅ (being, as a consequence, K a scattered compact set), then C (K ) is isomorphic to c0 and, so, there exists an equivalent norm k · k such that R(C (K ), k · k) = 1. From this result and that in Section 2, we can easily deduce that any Banach space which can be continuously embedded in C (K ), K as above, can be renormed to satisfy the w-FPP. This is a strict improvement of the result in [12], because, as proved in [15], when K is a Ciesielski–Pol’s compact, then K (3) = ∅, but C (K ) cannot be continuously embedded in c0 (Γ ) for any set Γ . Finally in Section 3, we use the results in Section 2 and Theorem 14 in [16] to obtain a generic result for the w-FPP under renormings in C (K ). Using the approach in [17], for a Banach space X , we denote by P the set of all equivalent norms with the metric ρ(p, q) = sup{|p(x)− q(x)| : x ∈ B} where B is the unit ball of X . Then, P is an open subset of the complete metric space formed by all continuous seminorms, and, thus, a Baire space. We will use σ -porosity to state the notion of negligible sets. Recall that any σ -porous set (see definition in Section 3) is of Baire first category. In [16] we had proved the following: let X be a Banach space such that R(X ) < 2. Then, there exists a σ -porous set A of P such that for every q ∈ P \ A the space (X , q) satisfies the w-FPP. This leads us to deduce that almost all renormings of the space C (K ), K as above, (in the sense of porosity) satisfy the w-FPP. This result can be understood as a continuation of the results in [14]. Indeed, in this paper it was proved that C (K ) endowed with the supremum norm, K as above, satisfies the w-FPP. Now we can add that C (K ) satisfies the w-FPP for almost all norms which are equivalent to the supremum one. 2. A renorming with the w-FPP Definition 2.1. Let X be a Banach space. The coefficient R(X ) is defined by R(X ) = sup{lim inf kxn + xk : {xn } is weakly null with kxn k ≤ 1; lim kxn k = 1 , kxk = 1}. The following lemmas will become very important tools to prove the main theorem in this section. Lemma 2.2 (Goebel–Karlovitz’s Lemma [18,19]). Let K be a weakly compact convex subset of a Banach space X , and T : K → K a non-expansive mapping. Assume that K is minimal under these conditions and {xn } is an approximate fixed point sequence for T . Then, limn→∞ kxn − xk = diam(K ) for every x ∈ K . Lemma 2.3 ([12]). Let K be a weakly compact convex subset of a Banach space X , and T : K → K a non-expansive mapping. Assume that K is minimal under these conditions, diam (K ) = 1 and {xn } is an approximate fixed point sequence for T which is weakly null. Then, for every ε > 0 and t ∈ [0, 1], there exists a subsequence of {xn }, denoted again {xn }, and a sequence {zn } in K such that: (i) (ii) (iii) (iv)

{zn } is weakly convergent to a point z ∈ K . kzn k > 1 − ε for every n ∈ N. kzn − zm k ≤ t for every n, m ∈ N. lim supn kzn − xn k ≤ 1 − t.

Lemma 2.4. Let (Y , k·kY ) be a Banach space with R(Y ) < 2, {yn } a weakly null and y ∈ Y \{0}. Assume that 0 < α = limn kyn kY and 0 < β = kykY . Then



lim sup kyn + ykY ≤ c lim kyn kY + kykY n

n α,β R(Y )−1+max β α n o β α, 1+max β α

n

where c =

o

< 1.



T.D. Benavides, S. Phothi / Nonlinear Analysis 72 (2010) 1409–1416

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Proof. It is clear that c < 1. Assume that α > β . Then we obtain that

 

yn y β

lim kyn kY β lim sup + + 1 − n α β Y α n β R(Y ) + α − β β(R(Y ) − 1) + α β(R(Y ) − 1) + α (α + β) α+β

lim sup kyn + ykY ≤ n

≤ = =

R(Y ) − 1 + βα

=

1 + βα

(α + β).

(2.1)

Slight modification of the above argument shows that, if β ≥ α , we obtain that β

R(Y ) − 1 + α (α + β). β 1+ α

lim sup kyn + ykY ≤ n

It follows that





lim sup kyn + ykY ≤ c lim kyn kY + kykY .  n

n

Theorem 2.5. Let (X , k · kX ) and (Y , k · kY ) be Banach spaces. Assume that R(Y ) < 2 and there exists a one-to-one linear continuous mapping J : X → Y . Then there exists an equivalent norm in X such that X endowed with the new norm satisfies the w-FPP. Proof. Define, for each x ∈ X ,

|x|2 = kxk2X + kJxk2Y . It is not difficult to check that | · | is an equivalent norm on X . We will show that the space (X , | · |) enjoys the w-FPP. Assume that C is a weakly compact convex subset of X and T : C → C is a | · |-non-expansive mapping. By Zorn’s lemma, there exists a convex closed subset K of X which is T -invariant and minimal under these conditions. This set must be separable (see [5], page 35–36) and each point of K is diametral. We will assume by contradiction that K is not singleton. Then by multiplication, we can assume that | · | − diam K = 1 and we can also assume that there exists a weakly convergent approximated fixed point sequence {xn } for T in K . By translation, we can assume that {xn } is weakly null and so, 0 ∈ K . By Goebel–Karlovitz’ lemma, limn |xn | = limn |xn − 0| = diam K = 1. Due to the separability of K , we can assume that limn |xn − x|, limn kxn − xkX and limn kJ (xn − x)kY do exist for every x ∈ K . We claim that limn kJxn kY > 0. To see this, assume by contradiction that limn kJxn kY = 0. Since diam K = 1 and J is a one-to-one linear mapping, we can choose x in K such that kJxkY = a > 0. According to Goebel–Karlovitz’ lemma, 1 = lim |xn − x|2 n

= lim kxn − xk2X + kJ (xn − x)k2Y




n

≥ lim kxn − xk2X + a2 .

(2.2)

lim kxn − xk2X ≤ 1 − a2 .

(2.3)

n

Thus n

By (2.3), we obtain that

lim xn − n



= lim xn −

x 2

2

n

≤ =

1 4 1



 x 

2

+ J xn −

x 2

2

2

X

Y

lim (kxn − xkX + kxn kX ) + (kJ (xn − x)kY + kJxn kY )2 2




n

lim |xn − x|2 + |xn |2 + 2(kxn − xkX kxn kX + kJ (xn − x)kY kJxn kY )

4 n  p 1 ≤ 1 + 1 + 2 1 − a2 4 <1

which contradicts to Goebel–Karlovitz’ lemma since Thus limn kJxn kY > 0.

x 2

belongs to K .




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Assume that limn kJxn kY = 3b for some positive real number b. Choose 0 < γ < R(Y )−1+ˆc 1+ˆc

√

b(1−b− 1−2b) 2

and let cˆ =

1−b

γ

. Denote

c= < 1. We apply Lemma 2.3 for t = 1 − b and

(  < min

γ2 2

(1 − c ), 2

b( 1 − b −

√

1 − 2b)

!

2

to obtain a subsequence of {xn }, denoted again by {xn }, and a sequence {zn } in K which satisfy (i)–(iv). Assume that z = w − lim zn . WOLOG we can assume that all real sequences below ({|zn − xn |}, kJ (xn − zn )k, etc) are convergent. By (iv) and the weak lower semi-continuity of the norm, we have

|z | ≤ lim inf |zn − xn | ≤ 1 − t = b

(2.4)

n

and by (iii), lim |zn − z | ≤ lim lim |zn − zm | ≤ t = 1 − b. n

n

(2.5)

m

Moreover, again by using (iv), we obtain b2 ≥ lim sup |xn − zn |2 ≥ lim sup kJ (xn − zn )k2Y . n

n

Thus b ≥ lim sup kJ (xn − zn )kY ≥ lim sup (kJxn kY − kJzn kY ) n

n

which implies lim kJzn kY ≥ lim kJxn kY − b = 2b. n

(2.6)

n

We will split the proof into two cases: Case I. Assume that kJz kY ≤ γ . In this case, we have, by (2.6) lim kJ (zn − z )kY ≥ lim kJzn kY − kJz kY ≥ 2b − γ . n

n

By (2.5), we have

(1 − b)2 ≥ lim |zn − z |2 n

= lim kzn − z k2X + kJ (zn − z )k2Y




n

≥ lim kzn − z k2X + (2b − γ )2 . n

Since γ < b we have lim kzn − z k2X ≤ (1 − b)2 − (2b − γ )2 < (1 − b)2 − b2 = 1 − 2b. n

Thus, by using (ii), (2.4), (2.5) and (2.7), we obtain the following contradiction:

(1 − )2 ≤ lim |zn |2 n

= lim kzn k2X + kJzn k2Y




n

≤ lim (kzn − z kX + kz kX )2 + (kJ (zn − z )kY + kJz kY )2




n


= lim |zn − z |2 + |z |2 + 2(kzn − z kX kz kX + kJ (zn − z )kY kJz kY ) n √ ≤ (1 − b)2 + b2 + 2b 1 − 2b + 2γ √ √ ≤ (1 − b)2 + b2 + 2b 1 − 2b + b(1 − b − 1 − 2b) √ √ √ ≤ (1 − b)2 + b2 + 2b 1 − 2b + 2b(1 − b − 1 − 2b) − b(1 − b − 1 − 2b) < 1 − 2.

(2.7)

T.D. Benavides, S. Phothi / Nonlinear Analysis 72 (2010) 1409–1416

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Case II. Assume that kJz kY ≥ γ . By the assumption, we have 0 < γ ≤ kJz kY ≤ |z | ≤ b. Furthermore, lim kJ (zn − z )kY ≥ lim kJzn kY − kJz kY ≥ 2b − b = b. n

n

Hence 0 < b ≤ lim kJ (zn − z )kY ≤ lim |zn − z | ≤ 1 − b. n

n

Applying Lemma 2.4 with {yn } = {J (zn − z )} and y = Jz and using that the inequality R(Y ) < 2 implies that the function R(Y )−1+t t 7→ is increasing on the interval [0, +∞), we obtain that 1 +t lim kJ (zn − z ) + Jz kY ≤ n

 R(Y ) − 1 + cˆ  lim kJ (zn − z )kY + kJz kY n 1 + cˆ

  = c lim kJ (zn − z )kY + kJz kY .

(2.8)

n

Then by (ii), and (2.8), we have

(1 − )2 ≤ lim |zn |2 n

= lim kzn k2X + kJzn k2Y




n


≤ lim (kzn − z kX + kz kX )2 + kJ (zn − z ) + Jz k2Y n  2 ≤ lim (kzn − z kX + kz kX )2 + c 2 lim kJ (zn − z )kY + kJz kY n

n

 2
= lim (kzn − z kX + kz kX )2 + (kJ (zn − z )kY + kJz kY )2 + (c 2 − 1) lim kJ (zn − z )kY + kJz kY n n
2 2 ≤ lim |zn − z | + |z | + 2(kzn − z kX kz kX + kJ (zn − z )kY kJz kY ) − (1 − c 2 )kJz k2Y . n

(2.9)

Assume that limn |zn − z | = u and |z | = v . Denote t = limn kzn − z kX and s = kz kX . Then we have



lim kJ (zn − z )kY = lim |zn − z |2 − lim kzn − z k2X n

n

 21

=

p

n

u2 − t 2

and

kJz kY = |z |2 − kz k2X


12

=

p

v 2 − s2 .

Consider the function of two variables

p

f (t , s) = 2ts + 2 u2 − t 2

p v 2 − s2 .

By elementary calculus, we obtain that max

[0,u]×[0,v]

f (t , s) = 2uv

hence



2 kz kX lim kzn − z kX + kJz kY lim kJ (zn − z )kY n



n

≤ 2|z | lim |zn − z | n

and by using (2.4), (2.5), (2.9) becomes


(1 − )2 ≤ lim |zn − z |2 + |z |2 + 2|zn − z ||z | − (1 − c 2 )kJz k2Y n

≤ (1 − b)2 + b2 + 2b(1 − b) − (1 − c 2 )γ 2 = 1 − (1 − c 2 )γ 2 < 1 − 2 and we again reach the contradiction. Therefore the space (X , | · |) enjoys the w-FPP.



(2.10)

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3. A renorming in space of continuous functions If A is a subset of a topological space M, the derived set of A is the set A(1) of all accumulation points of A. If α is an ordinal number, we define the α th -derived set by transfinite induction: A(0) = A

A(α+1) = (A(α) )(1)

A(λ) =

\

A(α)

α<λ

where λ is a limit ordinal. We will need the following technical lemma: Lemma 3.1. Let A be the subset of Rn formed by all vectors x = (α1 , . . . , αn ) such that 0 ≤ αn ≤ · · · ≤ α1 , α12 + · · · + αn2 ≤ 1, and B the subset of Rn formed by all vectors y = (β1 , . . . , βn−1 , 0) such that 0 ≤ βn−1 ≤ · · · ≤ β1 , β12 + · · · + βn2−1 ≤ 1. Define φ : A × B → R by

φ(x, y) = max{α12 , (β1 + α2 )2 } + · · · + max{αn2−1 , (βn−1 + αn )2 } + αn2 . Then, max{φ(x, y) : x ∈ A, y ∈ B} ≤ 4 − n−1 . Proof. Since A × B is a compact subset of R2n and φ is a continuous function, we know that φ attains a maximum M at a point in A × B. We will check that M ≤ 4 − n−1 . First, we will prove that M ≤ 4 − n−1 if for some k ∈ {1, . . . , n − 1} we have max{αk , (βk + αk+1 )} = αk . Indeed, assume that for some k ∈ {1, . . . , n − 1} we have that max{αj , βj + αj+1 } = βj + αj+1 for j = 1, . . . , k − 1 and max{αk , βk + αk+1 } = αk . Thus,

φ(x, y) ≤ (β1 + α2 )2 + · · · + (βk−1 + αk )2 + αk2 + (βk+1 + αk+1 )2 + · · · + (βn−1 + αn−1 )2 + αn2 = ku + vk2 where u = (β1 , . . . , βk−1 , 0, βk+1 , . . . , βn−1 , 0) and v = (α2 , . . . , αk , αk , αk+1 , . . . , αn−1 , αn ) and k · k denotes the Euclidean norm. Since kvk2 = kxk2 + αk2 − α12 and ku − vk ≥ αk the parallelogram identity implies 1

ku + vk2 ≤ 2 + 2kxk2 + 2αk2 − 2α12 − αk2 ≤ 2 + 2kxk2 − α12 ≤ 4 − . n

Now, consider the case max{αk , βk + αk+1 } = βk + αk+1 for every k = 1, . . . , n − 1. Then φ(x, y) = (β1 + α2 )2 + · · · + (βn−1 + αn )2 + αn2 = ku + yk2 where u = (α2 , . . . , αn , αn ). Since ku − yk ≥ αn , the parallelogram identity gives us 1

ku + yk2 ≤ 2 + 2kuk2 − αn2 ≤ 2 + 2kxk2 + 2αn2 − 2α12 − αn2 ≤ 2 + 2kxk2 − α12 ≤ 4 − .  n

(m) Theorem 3.2. Assume = ∅. Then, there exists a norm k · k equivalent to the supremum norm k · k∞ such that √ that K R(C (K ), k · k) ≤ 4 − m−1 . 2 Proof. For any x ∈ C (K ) denote by αk = max |x(t )| : t ∈ K (k−1) and define kxk2 = α12 + · · · + αm . It is clear that this norm is equivalent to the supremum norm. Assume that x ∈ C (K ), kxk √ ≤ 1 and {xn } is a weakly null sequence in C (K ) such that kxn k ≤ 1 for all n ∈ N. We will prove that lim supn kx + xn k ≤ 4 − m−1 . Let ε be an arbitrary positive number. By induction, we will define some subsets of K depending on ε . Denote L0 = K . Since K (m−1) is a finite set, there exists an (m−1) open subset U1 of K , containing K (m−1) , such that |x(t )| ≤ αm + ε for t ∈ U 1 . Denote L1 = K \ U1 . We have L1 =∅



(m−2)

which implies that L1



(m−2)

is a finite set contained in K (m−2) . Thus, there exists an open set U2 , containing L1 (m−3)

(m−3)

such that

|x(t )| ≤ αm−1 + ε for every t ∈ U 2 . We define L2 = L1 \ U2 . Then, L2 is a finite subset of K . By induction, we can assume that we have defined open sets U1 , . . . , Uk and compact sets L1 , . . . , Lk such that for j = 1, . . . , k we have (m−j−1) (m−j) (m−k−1) Lj = Lj−1 \ Uj , Lj is a finite subset of K (m−j−1) , Lj−1 ⊂ Uj and |x(t )| ≤ αm−j+1 + ε for every t ∈ U j . Since Lk is a ( m − k − 1 ) finite subset of K (m−k−1) , there exists an open set Uk+1 , containing Lk such that |x(t )| ≤ αm−k + ε for every t ∈ U k+1 . (m−k−1) We have that Lk+1 ⊂ L(km−k−1) \ Uk+1 = ∅, which implies that Lk(m+−1 k−2) is a finite set. Thus, we can construct open sets (m−j−1) is a finite subset U1 , . . . , Um−1 and compact sets L1 , . . . , Lm−1 such that for j = 1, . . . , m − 1 we have Lj = Lj−1 \ Uj , Lj (m−m) (m−j−1) (m−j) of K , Lj−1 ⊂ Uj and |x(t )| ≤ αm−j+1 + ε for every t ∈ U j . Moreover, Lm−1 = Lm−1 is a finite set. Since the sequence {xn } is weakly null, we can assume that |xn (t )| < ε for every t ∈ Lj(m−j−1) for j = 0, . . . , m − 1 and n large enough. Denote  βj = βj (n) = max |xn (t )| : t ∈ K (j−1) . We claim that kx + xn k2 ≤ φ(x, y) + O(ε), where φ is the function in Lemma 3.1 and O(ε) → 0 as ε → 0. Since (A ∪ B)0 = A0 ∪ B0 and K ⊂ Lj ∪ U1 ∪ ... ∪ Uj for j = 1, . . . , m − 1, we have (m−j−1)

K (m−j−1) ⊂ Lj

j [[ i =1

Ui .

T.D. Benavides, S. Phothi / Nonlinear Analysis 72 (2010) 1409–1416

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 We should compute max |(x + xn )(t )| : t ∈ K (m−j−1) for j = 1, . . . , m − 1. We have two possibilities: Sj (1) Assume that t ∈ i=1 Ui . In this case we have that |x(t )| ≤ αm−j+1 + ε and so, |x(t ) + xn (t )| ≤ αm−j+1 + βm−j + ε . Sj (m−j−1) (2) Assume that t ∈ K (m−j−1) \ i=1 Ui . In this case t ∈ Lj which implies that |xn (t )| ≤ ε and |x(t )+xn (t )| ≤ αm−j +ε . Thus,

max |x(t ) + xn (t )| : t ∈ K (m−j−1) ≤ max αm−j+1 + βm−j , αm−j + ε









for j = 1, . . . , m − 1 and max |x(t ) + xn (t )| : t ∈ K (m−1) ≤ αm + ε.





Hence,

kx + xn k2 ≤ max{(α1 + ε)2 , (β1 + α2 + ε)2 } + · · · + max{(αm−1 + ε)2 , (βm−1 + αm + ε)2 } + (αm + ε)2 ≤ φ(x, y) + O(ε) √ where O(ε) tends to 0 as ε → 0. Thus, by Lemma 3.1 we have that lim supn kx + xn k ≤ 4 − m−1 + O(ε). Since ε is arbitrary we easily obtain the conclusion.



Assume that Γ is a uncountable set. We can assume that Γ is endowed with the discrete topology. Let K be the one-point compactification of Γ . Then, (c0 (Γ ), k · k∞ ) is isometrically contained in (C (K ), k · k∞ ) and K (2) = ∅. From, Lemma 2.4 we obtain the following result which strictly improves the main result in [12], where it is proved that any Banach space which can be continuously embedded in c0 (Γ ) has an equivalent norm with the w-FPP. Corollary 3.3. Let X be a Banach space which can be continuously embedded in (C (K ), k · k∞ ) for some compact set K such that K (ω) = ∅. Then, X can be renormed to satisfy the w-FPP. Note that this is a strict improvement of the main result in [12], because, as proved in [15], when K is a Ciesielski–Pol’s compact, then K (3) = ∅, but C (K ) cannot be continuously embedded in c0 (Γ ) for any set Γ . 4. Genericity of the w-FPP in space of continuous functions Following the approach in [17] (see also [20]), for a Banach space (X , k · k), with closed unit ball B, we denote by P the Baire space of all equivalent norms with the metric ρ(p, q) = sup{|p(x) − q(x)| : x ∈ B}. In a Baire space, we can consider first category sets as negligible sets. However, there are some other different notions to ‘‘measure’’ the size of a set. For instance, in a measure space (X , µ, Σ ), a set A ⊂ X can be considered small if µ(A) = 0. However, in the real line, null measure is not equivalent to Baire first category. In fact, the real line can be decomposed into a disjoint union of two small sets: a null set and a first category set. To avoid this problem, we can consider a deeper notion of negligible set: Definition 4.1. Let M be a metric space. A subset A of M is said to be porous if there exist 0 < β ≤ 1 and r0 > 0 such that, for every x ∈ A and 0 < r ≤ r0 there exists y ∈ X such that B(y, β r ) ⊂ B(x, r ) ∩ (M \ A). A subset A of M is called σ -porous if A is the union of a countable family of porous sets. Porous and σ -porous sets can be considered ‘‘small’’ in X . In particular a σ -porous set is of Baire first category and, for X = Rn , a σ -porous set is a null set with respect to the Lebesgue measure. In [16] (Theorem 14), it is proved that if X is a Banach space such that R(X ) < 2, then there exists a σ -porous set A ⊂ P such that if q ∈ P \ A the space (X , q) satisfies the w-FPP. From this and Theorem 3.2 we easily obtain the following generic result: Corollary 4.2. Assume that K (ω) = ∅ and P is the set of all norms in C (K ) which are equivalent to the supremum norm with the metric ρ(p, q) = sup{|p(x) − q(x)| : x ∈ B}. Then, there exists a σ -porous set A ⊂ P such that if q ∈ P \ A the space (C (K ), q) satisfies the w-FPP. Acknowledgements The first author is partially supported by DGES, Grant BFM 2006-13997-C02-01 and Junta de Andalucía, Grant FQM127 and the second author is partially supported by Junta de Andalucía, Grant FQM-127 and The Commission on Higher Education, Thailand. References [1] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow, 1993. [2] M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucía, J. Pelant, V. Zizler, Functional Analysis and Infinite-dimensional Geometry, in: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001.

1416 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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