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Locally Lipschitz Functions and Bornological

arXiv:math/9302211v1 [math.FA] 9 Feb 1993Locally Lipschitz Functions and BornologicalDerivativesJ.M. Borwein1Department of Combinatorics and OptimizationUniversity of Waterloo, Waterloo, Ontario N2L 3G1M.

Fabian 2Department of MathematicsMiami University, Oxford, Ohio 45056J. Vanderwerff3Department of Pure MathematicsUniversity of Waterloo, Waterloo, Ontario N2L 3G1February 22, 2018AbstractWe study the relationships between Gateaux, Weak Hadamard and Fr´echet differentiabilityand their bornologies for Lipschitz and for convex functions.AMS Subject Classification.

Primary: 46A17, 46G05, 58C20.Secondary: 46B20.Key Words: Gateaux differentiability, Fr´echet differentiability, weak Hadamard differentiability,convex functions, Lipschitz functions, bornologies, Dunford-Pettis property, Grothendieck property,Schur property, not containing ℓ1.1Research supported in part by an NSERC (Canada) operating grant.2Research completed while visiting the Faculty of Mathematics, University of Waterloo.3NSERC (Canada) postdoctoral fellow.1

1IntroductionWe begin by introducing the definitions and notation that will be used. Unless otherwise specified,X is an infinite dimensional real Banach space with norm ∥· ∥and dual space X∗.

A bornologyB on X is a family of bounded subsets of X such that ∪{B : B ∈B} = X. We will focus on thefollowing bornologies: G = {singletons}, H = {compact sets}, WH = {weakly compact sets} andF = {bounded sets}.

Observe that G ⊂H ⊂WH ⊂F.A function f : X →R is called B-differentiable at x ∈X if there is Λ ∈X∗such that for eachB ∈B,1thf(x + th) −f(x) −⟨Λ, th⟩i→0 as t ↓0uniformly for h ∈B. Let F denote a family of real-valued locally Lipschitz functions on X; wewill usually consider locally Lipschitz (loclip), Lipschitz (lip), distance (dist), continuous convex(conv) and norms (norm); it is, of course, easy to check that continuous convex functions are locallyLipschitz ([10], Proposition 1.6).

For two bornologies on a fixed Banach space X, say F and Gand a family of functions F, we will write FF = GF, if for every f ∈F and every x ∈X, f isF-differentiable at x if and only if f is G-differentiable at x. Since Gloclip = Hloclip, we will writeG and H interchangeably.In the paper [1], it was shown that Hconv = Fconv if and only if X is infinite dimensional.From this, one might be tempted to believe that various notions of differentiability for convexfunctions coincide precisely when the bornologies on the space coincide.

However, this is far frombeing the case; for example, according to ([1], Theorem 2), WHconv = Fconv if and only if X ̸⊃ℓ1.In contrast to this, we will show in Section 2 that differentiability notions coincide for Lipschitzfunctions precisely when the bornologies are the same (for the H, WH and F bornologies). In thethird section we will study the relationship between WH-differentiability and H-differentiabilityfor continuous convex functions.

In particular, we show that if BX∗is w∗-sequentially compact,then Hconv = WHconv precisely when H = WH. However, there are spaces for which Hconv =WHconv and yet H ̸= WH.

This leads to examples showing that one cannot always extend aconvex function from a space to a superspace while preserving G-differentiability at a prescribedpoint. Some characterizations of the Schur and Dunford-Pettis properties are also obtained in termsof differentiabilty of continuous w∗-lower semicontinuous convex functions on the dual space.2Lipschitz functions and bornologiesAs mentioned in the introduction, there are spaces for which WH ̸= F but WHconv = Fconv.However, this is not the case for Lipschitz functions.Theorem 2.1 For a Banach space X, the following are equivalent.

(a) X is reflexive. (b) WHlip = Flip.2

(c) WHdist = Fdist.In order to prove this theorem, we will need a special type of sequence in nonreflexive Banachspaces. Namely, we will say {xk}∞k=1 ⊂X is a special sequence if there is an ǫ > 0 such that{zk} ⊂X has no weakly convergent subsequence whenever ∥xk −zk∥< ǫ.Remark (a) There are examples of sequences {xk} such that {xk} has no weakly convergentsubsequence but {xk} is not special.Indeed, let X = ℓ1 and consider yn,m = en + 1nem for m, n ∈N, ⋗> ⋉.

Let {xk} be anysequential arrangement of {yn,m}. It is not hard to verify {xk} has the desired properties.

Anotherexample is X = c0 and yn,m =nXk=1ek +n+mXk=n+11nek. (b) If f is Lipschitz and WH-differentiable at 0 (with f ′(0) = 0) but f is not Fr´echet differen-tiable at 0, it is not hard to construct a special sequence {xk}.Indeed, because f is not Fr´echet differentiable, we can choose {xk} in the unit sphere SX of Xand tk ↓0 which satisfy|f(tkxk) −f(0)|tk≥ǫfor some ǫ > 0.Using the fact that f is Lipschitz and WH-differentiable at 0, one can easily show that {xk} isspecial.□Part (b) of the above remark shows that in order to prove Theorem 2.1, it is necessary to showeach nonreflexive Banach space has a special sequence.

On the other hand, part (a) shows thatsuch sequences must be chosen carefully.Lemma 2.2 Suppose {xn} ⊂X has no weakly convergent subsequence. Then some subsequence of{xn} is a special sequence.Proof.If some subsequence of {xn} is special, then there is nothing more to do.

So we willsuppose this is not so and arrive at a contradiction by producing a weakly convergent subsequenceof {xn}.Given ǫ = 1, by our supposition, we choose N1 ⊂N and {z1,i}i∈N1 such that∥xi −z1,i∥< 1fori ∈N1andw−limi∈N1 z1,i = z1.Supposing Nk−1 has been chosen, we choose Nk ⊂Nk−1 and {zk,i}i∈Nk ⊂X satisfying∥xi −zk,i∥< 1kfor i ∈Nkandw−limi∈Nkzk,i = zk. (2.1)In this manner we construct {zk,i}i∈Nk and Nk for all k ∈N.Notice that zn −zm = w−limi∈Nn(zn,i −zm,i) for n > m. Thus by the w-lower semicontinuity of∥· ∥and (2.1) we obtain∥zn −zm∥≤lim infi∈Nn ∥zn,i −zm,i∥≤lim infi∈Nn (∥zn,i −xi∥+ ∥xi −zm,i∥) ≤1n + 1m ≤2m.3

Thus zn converges in norm to some z∞∈X.Now for each n ∈N choose integers in ∈Nn with in > n. We will show xinw→z∞. So letΛ ∈BX∗and ǫ > 0 be given.

We select an n0 ∈N which satisfies1n0< ǫ3and∥zm −z∞∥< ǫ3form ≥n0. (2.2)Because zn0,iw→zn0, we can select m0 so that|⟨Λ, zn0,i −zn0⟩| < ǫ3for alli ≥m0.

(2.3)For m ≥max {n0, m0}, we have|⟨Λ, xim −z∞⟩|≤|⟨Λ, xim −zn0,im⟩| + |⟨Λ, zn0,im −zn0⟩| + |⟨Λ, zn0 −z∞⟩|< ∥xim −zn0,im∥+ ǫ3 + ∥zn0 −z∞∥[by (2.3) since im > m ≥m0]<1n0 + ǫ3 + ǫ3 < ǫ. [by (2.2) and (2.1)]Therefore xinw→z∞.□Proof of Theorem 2.1.

Notice that (a) =⇒(b) =⇒(c) is trivial. It remains to prove (c) =⇒(a).Suppose X is not reflexive, hence BX is not weakly compact and so there exists {xn} ⊂SX with noweakly convergent subsequence.

By Lemma 2.2 there is a subsequence, again denoted by {xn}, anda ∆∈(0, 1) such that {zn} ⊂X has no weakly convergent subsequence whenever ∥zn −xn∥< ∆.By passing to another subsequence if necessary we may assume ∥xn −xm∥> δ for all n ̸= m, withsome 0 < δ < 1.For n = 1, 2, . .

. , let Bn = {x ∈X : ∥x −4−nxn∥≤δ∆4−n−1} and put C = X\∪∞n=1Bn.Because 4−m +δ∆4−m−1 < 4−n −δ∆4−n−1 for m > n, we have that Bn ∩Bm = ∅whenever n ̸= m.For x ∈X, let f(x) be the distance of x from C. Thus f is a Lipschitz function on X with f(0) = 0.We will check that f is WH-differentiable at 0 but not F-differentiable at 0.Let us first observe that f is G-differentiable at 0.

So fix any h ∈X with ∥h∥= 1. Then[0, +∞)h meets at most one ball Bn.

In fact assume tm, tn > 0 are such that ∥tih−4−ixi∥< δ∆4−i−1for i = n, m. Then |4iti −1| < δ∆4 for i = n, m and∥xn −xm∥≤∥xn −4ntnh∥+ ∥4ntnh −4mtmh∥+ ∥4mtmh −xm∥< δ∆4 + 2δ∆4+ δ∆4 = δ∆< δ.This means that n = m. It thus follows that for t > 0 small enough, we have f(th) = 0. Thereforef is G-differentiable at 0, with f ′(0) = 0.

Let us further check that f is not F-differentiable at 0.Indeed,f(4−nxn)∥4−nxn∥= δ∆4for all n,while ∥4−nxn∥→0.Finally assume that f is not WH-differentiable at 0. Then there are a weakly compact setK ⊂BX, ǫ > 0, and sequences {km} ⊂K, tm ↓0 such thatf(tmkm)tm> ǫfor all m ∈N.4

Hence, as f is 1-Lipschitz, we have inf ∥kn∥≥ǫ > 0. Further, because f(tmkm) > 0, there arenm ∈N such that∥tmkm −4−nmxnm∥< ∆δ4−mn−1,m = 1, 2, .

. .

.Consequently,∥4nmtmkm −xnm∥< ∆δ4 < ∆and |4nmtm∥km∥−1| < ∆δ4 . (2.4)Because {xn} is a special sequence with ∆, the first inequality in (2.4) says that {4nmtmkm} doesnot have a weakly convergent subsequence.

However the second inequality in (2.4) together withinf ∥kn∥> 0 ensures that 4nmtm is bounded and so, since {km} is weakly compact, 4nmtmkm has aweakly convergent subsequence, a contradiction. This proves f is WH-differentiable at 0.□Recall that a Banach space has the Schur property if H = WH, that is, weakly convergentsequences are norm convergent.Theorem 2.3 For a Banach space X, the following are equivalent.

(a) X has the Schur property. (b) Hlip = WHlip.

(c) Hdist = WHdist.Proof. It is clear that (a) =⇒(b) =⇒(c), thus we prove (c) =⇒(a).

Suppose X is not Schur andchoose {xn} ⊂SX such that xnw→0 but ∥xn∦→0. Since {xn} is not relatively norm compact,we may assume by passing to a subsequence if necessary that ∥xi −xj∥> δ for some δ ∈(0, 1)whenever i ̸= j.As in the proof of Theorem 2.1, let Bn = {x ∈X : ∥x−4−nxn∥≤δ4−n−1}, C = X\∪∞n=1Bn andlet f(x) = d(x, C).

Now f(0) = 0 and the argument of Theorem 2.1, shows that f is G-differentiableat 0 with f ′(0) = 0. However,f(4−nxn)4−n= δ4for all n ∈N.Since {xn} ∪{0} is weakly compact, it follows that f is not WH-differentiable at 0.□Remark.

Using the technique from the proof of Theorem 2.1, one can also prove the followingstatement. If a nonreflexive Banach space X admits a Lipschitzian Ck-smooth bump function,then it admits a Lipschitz function which is Ck-smooth on X\ {0} , WH-differentiable at 0, but notF-differentiable at 0.

A corresponding remark holds for non-Schur spaces.3Differentiability properties of convex functionsWe begin by summarizing some known results. First recall that a Banach space X has the Dunford-Pettis property if ⟨x∗n, xn⟩→0 whenever xnw→0 and x∗nw→0.

For notational purposes we will say5

X has the DP ∗if ⟨x∗n, xn⟩→0 whenever x∗nw∗→0 and xnw→0; see ([4], p. 177) for more onthe Dunford-Pettis property. Note that a completely continuous operator takes weakly convergentsequences to norm convergent sequences.

The proof of the next result is essentially in [1].Theorem 3.1 ([1])(a) X does not contain a copy of ℓ1 if and only if WHconv = Fconv if and only if WHnorm =Fnorm if and only if each completely continuous linear T : X →c0 is compact. (b) X has the DP ∗if and only if Hconv = WHconv if and only if Hnorm = WHnorm if andonly if each continuous linear T : X →c0 is completely continuous.

(c) X is finite dimensional if and only if Gconv = Fconv if and only if Gnorm = Fnorm if andonly if each continuous linear T : X →c0 is compact.Proof. Let us mention that (a) is contained in ([1], Theorem 2) and (c) is from ([1], Theorem 1).Whereas (b) can be obtained by following the proofs of ([1], Proposition 1 and Theorem 1).□If WHconv ̸= Gconv, for example, we can be somewhat more precise.Proposition 3.2 Suppose WHconv ̸= Gconv on X.

Then there is a norm ||| · ||| on X such that||| · ||| is not WH-differentiable at x0 ̸= 0 but ||| · |||∗is strictly convex at Λ0 ∈X∗\{0} satisfying⟨Λ0, x0⟩= |||x0||| |||Λ0|||.Proof. Following the techniques of [1], one obtains a norm ∥·∥on X such that ∥·∥is G-differentiableat x0 ̸= 0 but ∥· ∥is not WH-differentiable at x0.

Now define ||| · ||| on X by|||x||| = (∥x∥2 + d2(x, R↶⊬))⊮⊭Clearly d2(·, R↶⊬) is F-differentiable at x0 and so it follows that ||| · ||| is G-differentiable at x0 but||| · ||| is not WH-differentiable at x0 because ∥· ∥is not. Suppose now that {xn} satisfies2|||xn|||2 + 2|||x0|||2 −|||xn + x0|||2 →0.

(3.1)Then by convexity one obtains∥xn∥→∥x0∥, d2(xn, R↶⊬) →⊭(↶⊬, R↶⊬) = ⊬.¿From this one easily sees that ∥xn −x0∥→0.Now take Λ0 ∈X∗such that |||Λ0||| = 1 and ⟨Λ0, x0⟩= |||x0|||. We show that ||| · ||| is strictlyconvex at Λ0.

Suppose that |||x∗||| = 1 and |||x∗+ Λ0||| = 2, then choose {xn} with |||xn||| = |||x0||| sothat⟨x∗+ Λ0, xn⟩→2|||x0|||. (3.2)Consequently ⟨Λ0, xn⟩→|||x0||| and thus ⟨Λ0, xn + x0⟩→2|||x0|||; |||xn + x0||| →2|||x0|||.

But then{xn} satisfies (3.1) and so ∥xn −x0∥→0. This with (3.2) shows that ⟨x∗, x0⟩= |||x0|||.

Because6

||| · ||| is G-differentiable at x0, we conclude that x∗= Λ0. This proves the strict convexity of ||| · |||at Λ0.□One can also formulate similar statements (and proofs) for the cases Gconv ̸= Fconv andWHconv ̸= Fconv.We now turn our attention to spaces for which WHconv = Hconv.

Let us recall that a Banachspace X has the Grothendieck property if w∗-convergent sequences in X∗are weakly convergent;see ([4], p. 179). The following corollary is an immediate consequence of Theorem 3.1 (b).Corollary 3.3 If X has the Dunford-Pettis property and the Grothendieck property, then WHconv =Hconv.In particular, note that ℓ∞has the Grothendieck property (cf [3], p.103) and the Dunford-Pettis property (cf [4], p.177).Thus, unlike the case for Lipschitz functions, one can haveWHconv = Hconv for non-Schur spaces.

It will follow from the next result that these non-Schurspaces must be quite large, though.Theorem 3.4 For a Banach space X, the following are equivalent. (i) X has the DP ∗(ii) Hconv = WHconv(iii) If BY ∗is w∗-sequentially compact, then any continuous linear T : X →Y is completelycontinuous.Proof.By Theorem 3.1(b) we know that (i) and (ii) are equivalent and that (iii) implies (i).We will show (i) implies (iii) by contraposition.

Suppose (iii) fails, that is, there is an operatorT : X →Y which is not completely continuous for some Y with BY ∗w∗-sequentially compact.Hence we choose {xn} ⊂X such that xnw→0 but ∥Txn∦→0. Because Txnw→0, we know that{Txn} is not relatively norm compact.

Hence letting En = span {yk : k ≤n} with yk = Txk weknow there is an ǫ > 0 such that supkd(yk, En) > ǫ for each n.By passing to a subsequence, if necessary, we assume d(yn, En−1) > ǫ for each n. Now chooseΛn ∈BY ∗such that ⟨Λn, x⟩= 0 for all x ∈En−1 and ⟨Λn, xn⟩≥ǫ. Because BY ∗is w∗-sequentiallycompact, there is a subsequence Λnk such that Λnkw∗→Λ ∈BY ∗.

Observe that ⟨Λn, yk⟩= 0 forn > k and consequently ⟨Λ, yk⟩= 0 for all k. Now let z∗k = T ∗(Λnk −Λ) and zk = xnk. Certainlyz∗kw∗→0 and zkw→0 while ⟨z∗k, zk⟩= ⟨Λnk −Λ, Txnk⟩= ⟨Λnk −Λ, ynk⟩≥ǫ for all k. This shows thatX fails the DP ∗.□Corollary 3.5 If X has a w∗-sequentially compact dual ball or, more generally, if every separablesubspace of X is a subspace of a complemented subspace with w∗-sequentially compact dual ball,then the following are equivalent.7

(a) WHconv = Hconv. (b) X has the Schur property.Proof.Note that (b) =⇒(a) is always true, so we show (a) =⇒(b).

If BX∗is w∗-sequentiallycompact and X is not Schur then I : X →X is not completely continuous and Theorem 3.4 applies.More generally, suppose xnw→0 but ∥xn∦→0 and span {xn} ⊂Y with BY ∗w∗-sequentiallycompact. If there is a projection P : X →Y , then P is not completely continuous since P|Y is theidentity on Y .□We can say more in the case that X is weakly countably determined (WCD); see [9] and ChapterVI of [2] for the definition and further properties of WCD spaces.Corollary 3.6 For a Banach space X, the following are equivalent.

(a) X is WCD and WHconv = Hconv(b) X is separable and has the Schur property.Proof. It is obvious that (b) =⇒(a), so we show (a) =⇒(b).

First, since BX∗is w∗-sequentiallycompact (see e.g. [9], Corollary 4.9 and [7], Theorem 11), it follows from Corollary 3.5 that X hasthe Schur property.

But WCD Schur spaces are separable (see e.g. [9], Theorem 4.3).□Remark(a) Corollary 3.5 is satisfied, for instance, by GDS spaces (see [7], Theorem 11) and spaces withcountably norming M-basis (see [11], Lemma 1).

Notice that ℓ1(Γ) has a countably normingM-basis for any Γ, thus spaces with countably norming M-bases and the Schur property neednot be separable. (b) If X∗satisfies WHconv = Fconv, then X also does (because L1 ⊂X∗if ℓ1 ⊂X (see [5]Proposition 4.2)) but not conversely (c0 and ℓ1); cf.

Theorem 3.1(a). (c) Let X be a space such that X is Schur but X∗does not have the Dunford-Pettis property (cf.

[4], p 178). Then X satisfies Hconv = WHconv but X∗does not satisfy Hconv = WHconv.

(d) There are spaces with the DP ∗that are neither Schur nor have the Grothendieck property;for example ℓ1 × ℓ∞. (e) It is well-known that ℓ∞has ℓ2 as a quotient ([8], p. 111).

Thus quotients of spaces with theDP ∗need not have the DP ∗. It is clear that superspaces of spaces with the DP ∗need nothave the DP ∗; the example c0 ⊂ℓ∞shows that subspaces need not inherit the DP ∗.

(f) Haydon ([6]) has constructed a nonreflexive Grothendieck C(K) space which does not containℓ∞. Using the continuum hypothesis, Talagrand ([12]) constructed a nonreflexive GrothendieckC(K) space X such that ℓ∞is neither a subspace nor a quotient of X.

Since C(K) spaceshave the Dunford-Pettis property (see [3], p. 113), both these spaces have the DP ∗.8

As a byproduct of Corollaries 3.3 and 3.5 we obtain the following example which is related toresults from ([13]).Example. Let X be a space with the Grothendieck and Dunford-Pettis properties such that X isnot Schur (e.g.

ℓ∞). Then there is a separable subspace Y (e.g.

c0) of X and a continuous convexfunction f on Y such that f is G-differentiable at 0 (as a function on Y ), but no continuous convexextension of f to X is G-differentiable at 0 (as a function on X); there also exist y0 ∈Y \{0} andan equivalent norm ∥· ∥on Y whose dual norm is strictly convex but no extension of ∥· ∥to X isG-differentiable at y0.Proof.Let Y be a separable non-Schur subspace of X. By Corollary 3.5, there is a continuousconvex function f on Y which is G-differentiable at 0, but is not WH-differentiable at 0.

Since anyextension ˜f of f also fails to be WH-differentiable at 0, it follows that ˜f is not G-differentiable at0 because X has the DP ∗. Because Y fails the DP ∗, there is a sequence {Λn} ⊂X∗such that Λnconverges w∗but not Mackey to 0.

By the proof of ([1], Theorem 3), there is a norm ∥· ∥on Ywhose dual is strictly convex that fails to be WH-differentiable at some y0 ∈Y \{0}; as above, noextension of ∥· ∥to X can be G-differentiable at y0.□We close this note by relating the Schur and Dunford-Pettis properties to some notions ofdifferentiability for dual functions.Theorem 3.7 For a Banach space X, the following are equivalent. (a) X has the Schur property.

(b) G-differentiability and F-differentiability coincide for w∗-ℓsc continuous convex functions onX∗. (c) G-differentiability and F-differentiability coincide for dual norms on X∗.

(d) Hlip = WHlip.Proof. Of course (a) and (d) are equivalent according to Theorem 2.3.

(a) =⇒(b): Suppose (b) does not hold. Then for some continuous convex w∗-ℓsc f on X∗,there exists Λ0 ∈X∗such that f is G-differentiable at Λ0 but f is not F-differentiable at Λ0.

Letf ′(Λ0) = x∗∗∈X∗∗. We also choose δ > 0 and K > 0 such that for x∗1, x∗2 ∈B(Λ0, δ) we have|f(x∗1) −f(x∗2)| ≤K∥x∗1 −x∗2∥(since f is locally Lipschitz).

Because f is not F-differentiable atΛ0, there exist tn ↓0, tn < δ2, Λn ∈SX∗and ǫ > 0 such thatf(Λ0 + tnΛn) −f(Λ0) −⟨x∗∗, tnΛn⟩≥ǫtn. (3.3)Because f is convex and w∗-ℓsc, using the separation theorem we can choose xn ∈X satisfying⟨xn, x∗⟩≤f(Λ0 + tnΛn + x∗) −f(Λ0 + tnΛn) + ǫtn2for all x∗∈X∗;(3.4)9

Putting x∗= −tnΛn in (3.4) and using (3.3) one obtains⟨xn, tnΛn⟩≥f(Λ0 + tnΛn) −f(Λ0) −ǫtn2≥⟨x∗∗, tnΛn⟩+ ǫtn2 .And hence, ∥xn −x∗∗∥≥ǫ2 for all n.Let η > 0 and fix x∗∈SX∗. Since f is G-differentiable at Λ0, there is a 0 < t0 < δ2 such thatfor |t| ≤t0 we have⟨x∗∗, tx∗⟩−f(Λ0 + tx∗) + f(Λ0) ≥−η2t0.

(3.5)Using (3.4) with the fact that f has Lipschitz constant K on B(Λ0, δ), for |t| ≤t0 we obtain⟨xn, tx∗⟩≤f(Λ0 + tnΛn + tx∗) −f(Λ0 + tnΛn) + ǫtn2≤f(Λ0 + tx∗) −f(Λ0) + ǫtn2 + 2Ktn.Choosing n0 so large that ǫtn2 + 2Ktn < η2t0 for n ≥n0, the above inequality yieldsf(Λ0 + tx∗) −f(Λ0) −⟨xn, tx∗⟩≥−η2t0 for n ≥n0, |t| ≤t0. (3.6)Adding (3.5) and (3.6) results in⟨x∗∗−xn, tx∗⟩≥−ηt0 for n ≥n0, |t| ≤t0.Hence |⟨x∗∗−xn, x∗⟩| ≤η for n ≥n0.

This shows that xnw∗→x∗∗. Combining this with the factthat ∥xn −x∗∗∦→0 shown above, we conclude that for some δ > 0 and some subsequence we have∥xni −xni+1∥> δ for all i.

However xni −xni+1w→0 (in X) because xni −xni+1w∗→0 (in X∗∗). Thisshows that X is not Schur.Since (b) =⇒(c) is obvious, we show that (c) =⇒(a).

Write X = Y × R and suppose thatX is not Schur. Then we can choose {yn} ⊂Y such that ynw→0 but ∥yn∥= 1 for all n. Let{γn} ⊂(12, 1) be such that γn ↑1 and define ||| · ||| on X∗= Y ∗× R by|||(Λ, t)||| = sup{|⟨Λ, yn⟩+ γnt|} ∨12(∥Λ∥+ |t|).This norm is dual since it is a supremum of w∗-ℓsc functions and the proof of ([1], Theorem 1)shows that ||| · ||| is Gateaux but not Fr´echet differentiable at (0, 1).□If X is not Schur, then the previous theorem ensures the existence of a w∗-ℓsc convex continuousfunction on X∗which is G-differentiable but not F-differentiable at some point.

The followingremark shows that we can be more precise if X ̸⊃ℓ1.Remark. If X ̸⊃ℓ1 and X is not reflexive, then there is a w∗-ℓsc convex f on X∗and Λ ∈X∗suchthat f is G-differentiable at Λ and f ′(Λ) ∈X∗∗\X (and, a fortiori, f is not Fr´echet differentiableat Λ).Proof.Let Y be a separable nonreflexive subspace of X.

Let y∗∈Y ∗be such that y∗doesnot attain its norm on BY . Let y∗∗∈SY ∗∗be such that ⟨y∗∗, y∗⟩= 1.

Note that y∗∗∈SY ∗∗\Y10

because y∗does not attain its norm on BY . By the Odell-Rosenthal theorem (see [3], p.236), choose{yn} ⊂BY such that ynw∗→y∗∗.

Now Y ∗∗= Y ⊥⊥⊂X∗∗and some careful “identification checks”show that ynw∗→y∗∗as elements of X∗∗and y∗∗∈X∗∗\X. Let Λ be a norm preserving extensionof y∗, then ⟨y∗∗, Λ⟩= 1 and we define f on X∗byf(x∗) = sup{⟨x∗, yn⟩−1 −an : n ∈N} where ⅁⋉↓⊬.We now show that y∗∗∈∂f(Λ).

Indeed,f(Λ + x∗) −f(Λ) = f(Λ + x∗)=supn{⟨Λ + x∗, yn⟩−1 −an}≥limn→∞{⟨Λ, yn⟩−1 −an + ⟨x∗, yn⟩} = ⟨y∗∗, x∗⟩.To see that f is G-differentiable, fix x∗∈X∗and let ǫ > 0. Choose n0 so that |⟨y∗∗−yn, x∗⟩| ≤ǫ∥x∗∥for n ≥n0.

Now if 2∥t x∗∥< min {a1, . .

. , an0}, we have0 ≤f(Λ + tx∗) −f(Λ) −⟨y∗∗, tx∗⟩=supn {⟨Λ + tx∗, yn⟩−1 −an} −⟨y∗∗, tx∗⟩=supn {⟨Λ, yn⟩−1 + ⟨yn −y∗∗, tx∗⟩−an}≤maxn0, supn≥n0{⟨yn −y∗∗, tx∗⟩−an}o≤supn≥n0{|⟨y∗∗−yn, tx∗⟩|} ≤ǫ∥tx∗∥.Thus f is G-differentiable at Λ with G-derivative y∗∗∈Y ∗∗\Y .□Using the results of [1] and ˘Smulyan’s test type arguments in a fashion similar to Theorem 3.7,one can also obtain the following result.

We will not provide the details.Theorem 3.8 For a Banach space X, the following are equivalent. (a) X has the Dunford-Pettis property.

(b) G-differentiability and WH-differentiability coincide for w∗-ℓsc, continuous convex functionson X∗. (c) G-differentiability and WH-differentiability coincide for dual norms on X∗.We next consider what happens for F a family of norms alone.Remark.

(a) Gnorm = Fnorm on X implies Gdualnorm = Fdualnorm on X∗, but not conversely. (b) Gnorm = WHnorm on X implies Gdualnorm = WHdualnorm, but not conversely.

(c) WHnorm = Fnorm on X does not imply WHdualnorm = Fdualnorm on X∗.11

Proof. (a) This is immediate from Theorem 3.1(c) and Theorem 3.7 (since there are Schur spacesthat are not finite dimensional).

(b) Since the DP ∗implies the Dunford-Pettis property the first part follows from Theorem3.1(b) and Theorem 3.8. However, if X is separable, then by Corollary 3.6, Hconv = WHconv ifand only if X is Schur.

Thus the separable space C[0, 1] does not satisfy Gnorm = WHnorm yet ithas the Dunford-Pettis property, and thus by Theorem 3.8 satisfies Gdualnorm = WHdualnorm. (c) On c0 one has WHnorm = Fnorm (see Theorem 3.1).

But ℓ1 = c∗0 is a separable dualspace and so it admits a dual G-norm (see [2], Theorem II.6.7(ii) and Corollary II.6.9(ii)). Thisnorm cannot be everywhere F-differentiable since ℓ1 is not reflexive (see [2], Proposition II.3.4).However, this dual norm is everywhere WH-differentiable since ℓ1 is Schur.

Thus we do not haveWHdualnorm = Fdualnorm on ℓ1.□In fact we can be more precise than we were in (c). Using Theorem 3.7, Theorem 3.8 andTheorem 3.1 along with results from [1] one can obtain the following chain of implications.X fails the Schur property but has the Dunford-Pettis property =⇒WHdualnorm ̸= Fdualnorm on X∗=⇒X fails the Schur property and X∗⊃ℓ1.References[1]J.M.

Borwein and M. Fabian, On convex functions having points of Gateaux differentiabilitywhich are not points of Fr´echet differentiability, preprint.[2]R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Long-man Monographs in Pure and Applied Mathematics (to appear).[3]J.

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