October 21, 1996Characterizations of Banach Spaces viaConvex and other Locally Lipschitz FunctionsJon Borwein1Department of Mathematics & StatisticsSimon Fraser UniversityBurnaby, B.C., Canada V5A 1S6Marian Fabian2Czech Academy of SciencesPrague, Czech RepublicJon Vanderwer�3Department of MathematicsWalla Walla CollegeCollege Place, WA 99324Abstract. Various properties of Banach spaces, including the re exivity and the Schurproperty of a space, are characterized in terms of properties of corresponding classes oflocally Lipschitz functions on those spaces.AMS Classi�cation. 46A55, 46B20, 52A41.1 Research partially supported by NSERC and the Shrum Endowment.2 Research partially supported by grants AV CR 101-95-02 and GA CR 201-94-0069.3 Research partially supported by a Walla Walla College Faculty Development Grant.

2 Borwein, Fabian & Vanderwerff1. Introduction: \Sequentially Re exive" spaces and motivating results. Wewill work with real Banach spaces and let BX and SX denote the closed unit ball andunit sphere of the Banach space X respectively. As in [2], we will say a Banach spaceis sequentially re exive if Mackey and norm convergence coincide sequentially in its dualspace. The following result was proved in [22]; see also [3, Theorem 5].Theorem 1.1. For a Banach space X, the following are equivalent.(a) X does not contain an isomorphic copy of `1.(b) X is sequentially re exive.Let us recall that a function f is Gateaux di�erentiable at x if there is a � 2 X� suchthat limt!0[f(x+ th)� f(x)� �(th)]=t = 0for each h 2 SX . If the above limit is uniform over weakly compact sets, then f isweak Hadamard di�erentiable at x; if the limit is uniform for h 2 BX , then f is Fr�echetdi�erentiable at x. We will denote these notions as G-di�erentiability, WH-di�erentiabilityand F-di�erentiability. In re exive spaces, it is clear that WH-di�erentiability and F-di�erentiability are the same because closed balls are weakly compact. Surprisingly, thesetwo notions of di�erentiability coincide for continuous convex functions in a much widerclass of spaces as the following theorem from [3] demonstrates.Theorem 1.2. For a Banach space X the following are equivalent.(a) X is sequentially re exive.(b) WH-di�erentiability and F-di�erentiability coincide for continuous convex functionson X.This result motivated us to consider the following question. In which Banach spaces doFr�echet di�erentiability and weak Hadamard di�erentiability coincide for general Lipschitzfunctions? The following observation adds to the intrigue of this question.Theorem 1.3. Suppose a sequentially re exive spaceX admits a LipschitzWH-di�erentiablebump function, then X is an Asplund space.Proof. By the smooth variation principle for bump functions (see e.g. [23, Theorem4.10]) every continuous convex function on X has a point of weak Hadamard di�erentia-bility. According to Theorem 1.2, every continuous convex function has a point of Fr�echetdi�erentiability. Thus X is an Apslund space. �In spite of the (misleading) evidence provided by Theorem 1.3 that WH-di�erentiabilityof Lipschitz bump functions on sequentially re exive Banach space has structural impli-cations similar to those of F-di�erentiable bump function, we will provide an elementaryproof of the following result in the next section.

Characterizations of Banach Spaces 3Theorem 1.4. For a Banach space X the following are equivalent.(a) X is re exive.(b) WH-di�erentiability and F-di�erentiability coincide for Lipschitz functions on X.Since we �rst proved Theorem 1.4, results have been established showing that WH-di�erentiability and F-di�erentiability are distinct notions on nonre exive spaces for func-tions much closer to convex functions than just Lipschitz functions (see Theorem 2.1).Nonetheless, these stronger results rely on signi�cantly deeper structural theorems in Ba-nach space theory, and as such, the essence as to why these notions of di�erentiability aredi�erent is not as apparent in their proofs.In addition to the implications of sequential re exivity in the study of di�erentiabilityof convex functions, it also has applications to other notions of fundamental importancein convex analysis as illustrated in the following result which comes from [5] and [7].Theorem 1.5. For a Banach space X the following are equivalent.(a) X is sequentially re exive.(b) Every continuous convex function bounded on weakly compact subsets ofX is boundedon bounded subsets of X.(c) If a sequence of lsc convex functions converges uniformly on weakly compact sets toa continuous a�ne function, then the convergence is uniform on bounded sets.Let us remark that there are several characterizations of Banach spaces not containing`1 in terms of functions that are weakly sequentially continuous. For a avour of results inthis direction, we are content to refer the reader to Guti�errez's paper [16] and the referencestherein.A major focus of this note is the natural question that arises from a comparison ofTheorems 1.2, 1.4 and 1.5. That is, do Theorems 1.2 and 1.5 remain valid for di�erencesof continuous convex functions? (These are variously called di�erence-convex functions ordc-functions.) Our aim is to provide an account of the main results of which we are awarethat address this and some related questions, including those pertaining to \one-sided"or \directional di�erentiability." We will say that a function f is directionally Frechetdi�erentiable at x if limt#0 [f(x+ th)� f(x)]=texists uniformly for h in BX . If the above limit is uniform over weakly compact sets, thenf is weak Hadamard directionally di�erentiable at x.In many respects, this paper is a much updated version of the unpublished manuscript[4], and as such presents many of the theorems and proofs of results therein. In thisrevision, we have endeavoured to state many related results obtained subsequent to ouroriginal version. Except in a few instances where we have included complete proofs that

4 Borwein, Fabian & Vanderwerffare surprisingly simple, we will be content to give references to the proofs of publishedresults.2. Characterizations of Re exive Spaces. We state the characterizations in termsof nonre exivity. One should also note that Theorem 2.1(c) and (e) show very sharplimitations on the possibilities of extending (b) in Theorem 1.2, whereas (b) and (d) showlimitations to possible extensions of (b) and (c) in Theorem 1.5.Theorem 2.1. For a Banach space X the following are equivalent.(a) X is not re exive.(b) There is a non-increasing sequence of equi-Lipschitz norms converging uniformly onweakly compact sets to a norm, but such that the convergence is not uniform onbounded sets.(c) There is a Lipschitz convex function � such that � is directionally weak Hadamarddi�erentiable at some point at which it is not directionally Fr�echet di�erentiable.(d) There are continuous convex functions f and h on X such that h� f is bounded onweakly compact sets, but unbounded on BX .(e) There are norms � and � on X such that � � � is weak Hadamard but not Fr�echetdi�erentiable at some point.(f) There is a Lipschitz function � such that � is weak Hadamard but not Fr�echet di�er-entiable at some point.(g) There is a sequence fxng � BX and a � > 0 such that any sequence fyng satisfyingkyn � xnk < � has no weakly convergent subsequence.Proof. The equivalence of (a) through (e) was established in [8, Theorem 1 and Corollary5]. We will present the elementary proof of the equivalence of (a), (f) and (g) from [4].(a) ) (g): Because X is not re exive, by the Eberlein-Smulian theorem there isa sequence fxng � BX with no weakly convergent subsequence. We will suppose nosubsequence of fxng satis�es the property in (g) and arrive at a contradiction by producinga weakly convergent subsequence of fxng.Given � = 1, by our supposition, we choose N1 � N and fz1;igi2N1 such that kxi �z1;ik < 1 for i 2 N1 and w-limi2N1 z1;i = z1. Supposing Nk�1 has been chosen, we chooseNk � Nk�1 and fzk;igi2Nk � X satisfyingkxi � zk;ik < 1k for i 2 Nk and w- limi2Nk zk;i = zk: (2:1)In this manner we construct fzk;igi2Nk and Nk for all k 2 N.

Characterizations of Banach Spaces 5Notice that zn � zm = w- limi2Nn(zn;i � zm;i) for n > m. Thus by the w-lower semicon-tinuity of k � k and (2.1) we obtainkzn � zmk � lim infi2Nn kzn;i � zm;ik � lim infi2Nn (kzn;i � xik+ kxi � zm;ik) � 1n + 1m � 2m:Thus zn converges in norm to some z1 2 X.Now for each n 2 N choose integers in 2 Nn with in > n. We will show xin w! z1.So let � 2 BX� and � > 0 be given. We select an n0 2 N which satis�es1n0 < �3 and kzm � z1k < �3 for m � n0: (2:2)Because zn0;i w! zn0 , we can select m0 so that��h�; zn0;i � zn0i�� < �3 for all i � m0: (2:3)For m � maxfn0;m0g, we have��h�; xim � z1i�� � ��h�; xim � zn0;imi��+ ��h�; zn0;im � zn0i��+ ��h�; zn0 � z1i��< kxim � zn0;imk+ �3 + kzn0 � z1k [by (2.3) since im > m � m0]< 1n0 + �3 + �3 < �: [by (2.2) and (2.1)]Therefore xin w! z1 contradicting the fact that fxng has no weakly convergent subse-quence.(g) ) (f): By (g) we choose fxng � BX , and a � 2 (0; 1) such that fzng � Xhas no weakly convergent subsequence whenever kzn � xnk < �. By passing to anothersubsequence if necessary we may assume kxn�xmk > � for all n 6= m, with some 0 < � < 1.For n = 1; 2; : : : ; let Bn = fx 2 X : kx � 4�nxnk � ��4�n�1g and put C =X n [1n=1Bn. Because 4�m + ��4�m�1 < 4�n � ��4�n�1 for m > n, we have thatBn \Bm = ; whenever n 6= m. For x 2 X, let f(x) be the distance of x from C. Thus fis a Lipschitz function on X with f(0) = 0. We will check that f is WH-di�erentiable at0 but not F-di�erentiable at 0.Let us now observe that f is G-di�erentiable at 0. Fix any h 2 X with khk = 1.Then [0;+1)h meets at most one ball Bn. In fact assume tm; tn > 0 are such thatktih� 4�ixik < ��4�i�1 for i = n;m. Then j4iti � 1j < ��4 for i = n;m andkxn � xmk � kxn � 4ntnhk + k4ntnh� 4mtmhk + k4mtmh� xmk< ��4 + 2��4 + ��4 = �� < �:

6 Borwein, Fabian & VanderwerffBecause kxn � xmk � � for m 6= n, this shows that n = m. It thus follows that for t > 0small enough, we have f(th) = 0. Therefore f is G-di�erentiable at 0, with f 0(0) = 0. Letus further check that f is not F-di�erentiable at 0. Indeed, f(4�nxn)k4�nxnk = ��4 for all n whilek4�nxnk ! 0.Finally assume that f is not WH-di�erentiable at 0. Then there are a weakly compactset K � BX ; � > 0, and sequences fkmg � K; tm # 0 such that f(tmkm)tm > � for all m 2 N.Hence, as f is 1-Lipschitz, we have inf kknk � � > 0. Further, because f(tmkm) > 0, thereare nm 2 N such that ktmkm � 4�nmxnmk < ��4�mn�1; m = 1; 2; : : : . Consequently,k4nmtmkm � xnmk < ��4 < � and j4nmtmkkmk � 1j < ��4 : (2:4)By our initial selection of fxng and �, the �rst inequality in (2.4) says that f4nmtmkmgdoes not have a weakly convergent subsequence. However the second inequality in (2.4)together with inf kknk > 0 ensures that 4nmtm is bounded. Now because fkmg is relativelyweakly compact, 4nmtmkm has a weakly convergent subsequence, a contradiction. Thisproves f is WH-di�erentiable at 0.(f) ) (a): This is clear since the unit ball cannot be weakly compact. �One should notice that it is also easy to derive (g) from (f) in the above theorem.Indeed, because f is not Fr�echet di�erentiable, we can choose fxkg in the unit sphere SXof X and tk # 0 which satisfyjf(tkxk)� f(0)jtk � � for some � > 0:Using the fact that f is Lipschitz and WH-di�erentiable at 0, one can easily show thatfxkg satis�es (g).We should also remark that the sequence as given in (g) has been used to characterizere exivity in terms of comparing two forms of set convergences of bounded weakly closedsets (see [6, Theorem 4.2(b)]). For a thorough account on set convergences and theirapplications, we recommend Beer's monograph [1].Unlike the proof presented above that (f) holds in nonre exive spaces, the proof givenin [8] that (b) through (e) are valid in nonre exive spaces uses a decidedly nonelementaryresult. Indeed, it relies on a result from [17] that uses Rosenthal's `1 theorem and hasthe Josefson-Nissenzweig theorem as a direct consequence. The following remark justi�esthe use of the Josefson-Nissenzweig theorem in the proof that (d) occurs in nonre exivespaces.

Characterizations of Banach Spaces 7Remark 2.2. The condition in Theorem 2.1(d) directly implies the existence of a sequencein SX� that converges weak� to 0.Proof. Since each continuous convex function is bounded below on BX , we know thatone of the functions, say h, is unbounded above on BX and so the remark follows from [5,Lemma 2.3]. We include a very simple proof of this. Let Fn = fx : h(x) � ng and choosexn 2 BX such that h(xn) > n. Using the separation theorem we choose �n 2 SX� suchthat �n(xn) � supFn �n. If �n did not converge weak� to 0 we could choose �x 2 X suchthat �n(�x) > 1 for in�nitely many n. However, �n(xn) � 1, and so �x 62 Fn for each nwhich means h(�x) > n for each n, a contradiction. �Sadly, we do not know if the Josefson-Nissensweig theorem can likewise be directlyderived from Theorem 2.1(b),(c) or (e).Notice that Theorem 2.1 illustrates that, with many properties, di�erences of convexfunctions have more in common with Lipschitz or locally Lipschitz functions than theydo with convex functions. Relatedly, however, it is not clear that there are any nonsuperre exive Banach spaces on which arbitrary Lipschitz functions can be approximateduniformly on bounded sets by di�erences of convex functions.While we have shown that WH-di�erentiability and F-di�erentiability are distinctnotions for Lipschitz functions (and hence for continuous convex functions), the implica-tions of the existence of WH-di�erentiable bump functions on sequentially re exive Banachspaces have not been completely clari�ed. In particular, we mention the following ques-tions arising from Theorem 1.3 (we refer the reader to the monograph [9] for an expositionon Banach spaces admitting smooth bump functions).Question 2.3. (a) If X is sequentially re exive and admits a Lipschitz WH-di�erentiablebump function, does X admit a Lipschitz F-di�erentiable bump function?(b) If X is sequentially re exive and admits a continuous WH-di�erentiable bumpfunction, is X an Asplund space?The existence of norms that are directionally Fr�echet di�erentiable has also receivedconsiderable attention; see for example [13], [15] and the references therein. The followingquestion, which is somewhat related to Question 2.3(b), was posed by Giles and Scif-fer in [14], where it was �rst shown that directional WH-di�erentiability need not implydirectional Fr�echet di�erentiability for convex functions on c0.Question 2.4. If the norm on X is directionally WH-di�erentiable and X 6� `1, is X anAsplund space?Concerning this question, let us note that [15] shows any space with a directionally F-di�erentiable norm is an Asplund space. However, Theorem 2.1 shows that directional WH-di�erentiability need not imply directional F-di�erentiability for convex functions on spaces

8 Borwein, Fabian & Vanderwerffthat do not contain `1. For some further questions related to directional di�erentiability,we refer the reader to [15].3. Spaces in whose duals weak� and Mackey convergence coincide sequentially.If X is such that weak� and Mackey convergence coincide sequentially in X�, we will sayX has the DP� property. Our interest in this property stems from the following theorem.Theorem 3.1. Let X be a Banach space. Then the following are equivalent.(a) X has the DP� property.(b) WH-di�erentiability and G-di�erentiability coincide for all lsc convex functions on X.(c) Every continuous convex function is bounded on weakly compact subsets of X.(d) A sequence of lsc convex functions ffng converges uniformly to a continuous a�nefunction � on weakly compact sets provided ffng converges pointwise to � on X.Proof. The equivalence of (a) and (b) was established in [3], while the equivalence of (a)and (c) is shown in [5]. The techniques of [5] were later used in [7] to show that (a) implies(d) (while (d) implies (a) is completely clear). �Let us recall some standard de�nitions. A Banach space is said to have the Schurproperty if weakly convergent sequences are norm convergence (equivalently weakly com-pact sets are norm compact). A Banach space X is said to have the Grothendieck propertyif weak� and weak convergence agree sequentially in X�. A Banach space X has theDunford-Pettis property if x�n(xn) ! 0 whenever x�n w�! 0 and xn w! 0. Notice that theDP� property di�ers from the Dunford-Pettis property only in that one merely requiresx�n w�! 0 rather than x�n w! 0 (and hence the notation `DP� property').Remark 3.2. (a) Every Banach space with the Schur property has the DP� property.(b) If X has the Dunford-Pettis property and the Grothendieck property, then X hasthe DP� property.(c) `1 is a space with the DP� property that does not have the Schur property.Proof. Both (a) and (b) follow directly from the de�nitions involved. For (c) recall that`1 has both the DP property (see [10, p. 103] and Grothendieck property (see [11, p.177]) but does not have the Schur property. �It is natural to ask when the Schur and DP� properties are equivalent. The nextresult and its corollaries will show that they are equivalent for relatively small spaces.First let us recall that a linear operator is said to be completely continuous if it mapsweakly convergent sequences to norm convergent sequences.

Characterizations of Banach Spaces 9Theorem 3.3. For a Banach space X, the following are equivalent.(a) X has the DP � property.(b) If BY � is weak�-sequentially compact, then each continuous linear T : X ! Y iscompletely continuous.Proof. (a) ) (b): We will prove this by contraposition. Suppose (b) fails, that is,there is an operator T : X ! Y which is not completely continuous for some Y withBY � weak�-sequentially compact. Hence we choose fxng � X such that xn w! 0 butkTxnk 6! 0. Because Txn w! 0, we know that fTxng is not relatively norm compact.Hence letting En = span fyk : k � ng with yk = Txk we know there is an � > 0 such thatsupk d(yk; En) > � for each n.By passing to a subsequence, if necessary, we assume d(yn; En�1) > � for each n. Nowchoose �n 2 BY � such that h�n; xi = 0 for all x 2 En�1 and h�n; xni � �. Because BY �is weak�-sequentially compact, there is a subsequence �nk such that �nk w�! � 2 BY � .Observe that h�n; yki = 0 for n > k and consequently h�; yki = 0 for all k. Now letz�k = T �(�nk � �) and zk = xnk . Certainly z�k w�! 0 and zk w! 0 while hz�k; zki = h�nk ��; Txnk i = h�nk � �; ynki � � for all k. This shows that X fails the DP � property.(b) ) (a): This follows, for example, from [3] which shows that weak Hadamard andGateaux di�erentiability agree for convex functions on X if each operator T : X ! c0 iscompletely continuous; and hence X has the DP� property in this case. �Corollary 3.4. If X has a weak�-sequentially compact dual ball or, more generally, ifevery separable subspace of X is a subspace of a complemented subspace with weak�-sequentially compact dual ball, then the following are equivalent.(a) X has the DP� property.(b) X has the Schur property.Proof. Because (b) ) (a) is true more generally, we show (a) ) (b). If BX� is weak�-sequentially compact andX is not Schur, then I : X ! X is not completely continuous andTheorem 3.3 applies. More generally, suppose xn w! 0 but kxnk 6! 0 and spanfxng � Ywith BY � weak�-sequentially compact. Supposing there is a projection P : X ! Y , thenP is not completely continuous since P jY is the identity on Y . �We can say more in the case that X is weakly countably determined (WCD); see [21]and [9, Chapter VI] for the de�nition and further properties of WCD spaces.Corollary 3.5. For a Banach space X, the following are equivalent.(a) X is WCD and has the DP� property.

10 Borwein, Fabian & Vanderwerff(b) X is separable and has the Schur property.Proof. It is obvious that (b) ) (a), so we prove (a) ) (b). First, since BX� is weak�-sequentially compact (see e.g. [21, Corollary 4.9] and [19, Theorem 11], it follows fromCorollary 3.4 that X has the Schur property. But WCD Schur spaces are separable (seee.g. [21, Theorem 4.3]). �Remark 3.6. The following items contain facts concerning the DP� property.(a) Corollary 3.4 is satis�ed, for instance, by Gateaux di�erentiability spaces|see [19, 23]for de�nition and [19, Theorem 11] for a proof that such spaces have weak� sequentiallycompact dual balls. A wide class of spaces for which every separable subspace is asubspace of a complemented separable subspace is the class of spaces with countablynormingM -basis; see [24, Lemma 1], see also [26] and the references therein for furtherproperties and de�nition of M-bases. Notice that `1(�) has a countably norming M -basis for any �, thus spaces with countably norming M -bases and the Schur propertyneed not be separable.(b) Let X be a space such that X has the Schur property but X� does not have theDunford-Pettis property (cf. [11, p. 178]). Then X has the DP� property, but X�does not have the DP� property.(c) There are spaces with the DP � property that have neither the Schur nor Grothendieckproperties; for example `1 � `1.(d) It is well-known that `1 has `2 as a quotient (see [20, p. 111]). Thus quotients ofspaces with the DP � need not have the DP �. It is clear that superspaces of spaceswith the DP � need not have the DP �; the example c0 � `1 shows that subspacesneed not inherit the DP �.(e) In [18], Haydon has constructed a nonre exive Grothendieck C(K) space that does notcontain `1. Using the continuum hypothesis, Talagrand constructed a nonre exiveGrothendieck C(K) space X such that `1 is neither a subspace nor a quotient of X(see [25]). Since C(K) spaces have the Dunford-Pettis property (see [10, p. 113]),both these spaces have the DP �.Given the previous corollaries and remark, it seems natural to ask what natural (widelystudied) properties of Banach spaces in conjunction with the DP� property imply that theBanach space has the Schur property. For example, as far as we know, the answer to thefollowing question is not known.Question 3.7. If a Banach space has an M -basis and the DP� property, does it have theSchur property?We close this section with a curious example concering the extension of convex func-tions where smoothness cannot be preserved.

Characterizations of Banach Spaces 11Example 3.8. Let X be a space with the Grothendieck and Dunford-Pettis propertiessuch thatX does not have the Schur property (e.g. `1). Then there is a separable subspaceY (e.g. c0) of X and a continuous convex function f on Y such that f is G-di�erentiableat 0 (as a function on Y ), but no continuous convex extension of f to X is G-di�erentiableat 0 (as a function on X); there also exist y0 2 Y nf0g and an equivalent norm k � k on Ywhose dual norm is strictly convex but no extension of k � k to X is G-di�erentiable at y0.Proof. Let Y be a separable non-Schur subspace of X. By Corollary 3.4, there is a con-tinuous convex function f on Y which is G-di�erentiable at 0, but is not WH-di�erentiableat 0. Since any extension ~f of f also fails to be WH-di�erentiable at 0, it follows that ~fis not G-di�erentiable at 0 because X has the DP �. Because Y fails the DP � property,there is a sequence f�ng � X� such that �n converges w� but not Mackey to 0. By theproof of [3, Theorem 3], there is a norm k � k on Y whose dual is strictly convex that failsto be WH-di�erentiable at some y0 2 Y nf0g; as above, no extension of k � k to X can beG-di�erentiable at y0. �For additional results concerning the extension of smooth norms the reader may wishto consult [27]; see also [9].4. Characterizations of spaces with Schur or Dunford-Pettis Properties. Thefollowing theorem provides some characterizations of Banach spaces with the Schur prop-erty.Theorem 4.1. For a Banach space X, the following are equivalent.(a) X has the Schur property.(b) G-di�erentiability and F -di�erentiability coincide for w�-lsc continuous convex func-tions on X�.(c) G-di�erentiability and F -di�erentiability coincide for dual norms on X�.(d) Each continuous weak�-lsc convex function on X� is bounded on bounded subsets ofX�.(e) G-di�erentiability and WH-di�erentiability agree for Lipschitz functions on X.(f) G-di�erentiability and WH-di�erentiability coincide for di�erences of Lipschitz convexfunctions.(g) Every continuous convex function on X is weak Hadamard directionally di�erentiable.Proof. Let us outline the equivalence of (a) and (e) �rst. Clearly (a) implies (e), so wewill show (e) implies (a).(e)) (a): We suppose X is not Schur and proceed by contraposition. Let fxng � SXbe such that xn w! 0. Since fxng is not relatively norm compact, we may assume bypassing to a subsequence if necessary that kxi � xjk > � for some � 2 (0; 1) wheneveri 6= j.

12 Borwein, Fabian & VanderwerffAs in the proof of Theorem 2.1, let Bn = fx 2 X : kx � 4�nxnk � �4�n�1g, C =Xn[1n=1Bn and let f(x) = d(x;C). Now f(0) = 0 and the argument of Theorem 2.1,shows that f is G-di�erentiable at 0 with f 0(0) = 0. However,f(4�nxn)4�n = �4 for all n 2 N:Since fxng [ f0g is weakly compact, it follows that f is not WH-di�erentiable at 0.(a) , (d): This was outlined in the remark in [5, p. 67]; see also Remark 2.2 andTheorem 4.2.(a) ) (b): Suppose (b) does not hold. Then for some continuous convex weak�-lscf on X�, there exists �0 2 X� such that f is G-di�erentiable at �0 but f is not F-di�erentiable at �0. Let f 0(�0) = x�� 2 X��. We also choose � > 0 and K > 0 such thatfor x�1; x�2 2 B(�0; �) we have jf(x�1)� f(x�2)j � Kkx�1 � x�2k (since f is locally Lipschitz).Because f is not F-di�erentiable at �0, there exist tn # 0; tn < �2 ;�n 2 SX� and � > 0 suchthat f(�0 + tn�n)� f(�0)� hx��; tn�ni � �tn: (4:1)Because f is convex and weak�-lsc, using the separation theorem we can choose xn 2 Xsatisfying hxn; x�i � f(�0 + tn�n + x�)� f(�0 + tn�n) + �tn2 for all x� 2 X�; (4:2)Putting x� = �tn�n in (4.2) and using (4.1) one obtainshxn; tn�ni � f(�0 + tn�n)� f(�0)� �tn2� hx��; tn�ni+ �tn2 :And hence, kxn � x��k � �2 for all n.Let � > 0 and �x x� 2 SX� . Since f is G-di�erentiable at �0, there is a 0 < t0 < �2such that for jtj � t0 we havehx��; tx�i � f(�0 + tx�) + f(�0) � ��2 t0: (4:3)Using (4.2) with the fact that f has Lipschitz constant K on B(�0; �), for jtj � t0 weobtain hxn; tx�i � f(�0 + tn�n + tx�)� f(�0 + tn�n) + �tn2� f(�0 + tx�)� f(�0) + �tn2 + 2Ktn:

Characterizations of Banach Spaces 13Choosing n0 so large that �tn2 + 2Ktn < �2 t0 for n � n0, the above inequality yieldsf(�0 + tx�)� f(�0)� hxn; tx�i � ��2 t0 forn � n0; jtj � t0: (4:4)Adding (4.3) and (4.4) results inhx�� � xn; tx�i � ��t0 for n � n0; jtj � t0:Hence jhx�� � xn; x�ij � � for n � n0. This shows that xn w�! x��. Combining this withthe fact that kxn � x��k 6! 0 shown above, we conclude that for some � > 0 and somesubsequence we have kxni � xni+1k > � for all i. However xni � xni+1 w! 0 (in X) becausexni � xni+1 w�! 0 (in X��). This shows that X is not Schur.Since (b) ) (c) is obvious, so we show that (c)) (a). Write X = Y �R and supposethat X is not Schur. Then we can choose fyng � Y such that yn w! 0 but kynk = 1 for alln. Let f ng � ( 12 ; 1) be such that n " 1 and de�ne jjj � jjj on X� = Y � �R byjjj(�; t)jjj = sup�jh�; yni+ ntj _ 12(k�k+ jtj):This norm is dual since it is a supremum of weak�-lsc functions and the proof of [3,Theorem 1] shows that jjj � jjj is Gateaux but not Fr�echet di�erentiable at (0; 1).Finally, (a) implies (g) follows from Dini's monotone convergence theorem, while (a)implies (f) is clear from the de�nitions; [8, Proposition 8] shows that both (f) and (g) failin spaces without the Schur property. �The Schur property can also be characterized by comparing pointwise and uniformconvergence of equi-Lipschitz functions; see [7, Proposition 3.5].Our �nal theorem provides a characterization of the Dunford-Pettis property via func-tions on the dual space.Theorem 4.2. For a Banach space X, the following are equivalent.(a) X has the Dunford-Pettis property.(b) G-di�erentiability and WH-di�erentiability coincide for real-valued w�-lsc convexfunctions on X�.(c) G-di�erentiability and WH-di�erentiability coincide for dual norms on X�.(d) Each continuous weak�-lsc convex function on X� is bounded on weakly compactsubsets of X�.Proof. The equivalence of (a), (b) and (c) can be proved using results from [3] andSmulian's test in a fashion similar to Theorem 4.1.

14 Borwein, Fabian & Vanderwerff(a) ) (d): Although this is outlined in [5, p. 67], we will present a simpler proofwhich does not rely on a weak� variation of the technical lemma [5, Lemma 2.6].Suppose there is a continuous weak�-lsc function f that is unbounded on a weaklycompact setW . Choose x�n 2W such that f(x�n) > n and let Fn := fx� 2 X� : f(x�) � ng.Because Fn is weak� closed and convex, we use the separation theorem to choose xn 2 Xsuch that xn(x�n) � supFn �n: (4:5)As in Remark 2.2, it follows that xn w�! 0 in X�� and hence xn w! 0 in X. Because W isweakly compact, by passing to a subsequence we may assume x�n converges weakly. Nowthe Dunford-Pettis property implies x�n(xn)! 0. However, this cannot occur because (4.5)and the fact that there is a neighborhood V of 0 in X� and K > 0 such that f(x�) � Kfor all x� 2 V .(d) ) (a): This follows from [5, Lemma 2.1]. �References1. G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers,The Netherlands, 1993.2. J. Borwein, Asplund spaces are \sequentially re exive", CORR 91{14, University ofWaterloo, 1991.3. J. Borwein and M. Fabian, On convex functions having points of Gateaux di�erentia-bility which are not points of Fr�echet di�erentiability, Canad. J. Math. 45 (1993),1121{1134.4. J. Borwein, M. Fabian and J. Vanderwer�, Locally Lipschitz functions and bornologi-cal derivatives, CECM Research Report 93-012, 1993 (http://cecm.sfu.ca/preprints/).5. J. Borwein, S. Fitzpatrick and J. Vanderwer�, Examples of convex functions andclassi�cations of normed spaces, J. Convex Analysis 1 (1994), 61{73.6. J. Borwein and J. Vanderwer�, Futher arguments for slice convergence in nonre exivespaces, Set-Valued Analysis 2 (1994), 529{544.7. J. Borwein and J. Vanderwer�, Epigraphical and Uniform Convergence of ConvexFunctions, Trans. Amer. Math. Soc. 348 (1996), 1617{1631.8. J. Borwein and J. Vanderwer�, Convex functions on Banach spaces not containing `1,Canad. Math. Bull. (to appear).9. R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces,Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman,1993.10. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics92, Springer-Verlag, Berlin{New York{Tokyo, 1984.

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