Jean-Paul Penot A SHORT PROOF OF THE

DEMONSTRATIO MATHEMATICA Vol. XLIII No 3 2010

10.1515/dema-2013-0254

Jean-Paul Penot A SHORT PROOF OF THE SEPARABLE REDUCTION THEOREM

Abstract. We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

The separable reduction theorem is an important result of nonsmooth analysis. It enables to pass from fuzzy sum rules in spaces with smooth norms to fuzzy sum rules in general Asplund spaces. Asplund spaces form the appropriate setting for such approximate rules and for extremal principles ([3], [5], [4], [6], [7], [8], [9], [13], [14], [15]). The proof of that result is rather sophisticated and long (see [1, pp. 243–258], [14, pp. 183–221]). It is the purpose of this note to present a short proof. While the core of the proof uses arguments similar to the ones in the original proofs, the simplification stems from elementary results about convexification. Such results of independent interest are gathered in Section 1. The characterization of nonemptiness of the Fréchet subdifferential of a function is recalled in Section 2. The proof of the separable reduction theorem is the object of Section 3. The last section is devoted to known consequences of the theorem which may motivate the study and give an idea of its usefulness. 1. Useful facts from convex analysis We assume the reader has a basic knowledge of convex analysis. In particular, we assume some familiarity with the Fenchel-Moreau subdifferential : for a function f : X → R∞ := R ∪ {+∞} on a normed vector space X, finite Key words and phrases: Asplund spaces, convex analysis, Fréchet subdifferential, nonsmooth analysis, subdifferential, sum rules. 2000 Mathematics Subject Classification: 46B20, 46B99, 46T20. Unauthenticated Download Date | 5/29/17 4:11 PM

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at x ∈ X, it is defined by (1) ∂f (x) := ∂F M f (x) := {x∗ ∈ X ∗ : ∀x ∈ X f (x) ≥ f (x) + hx∗ , x − xi}. We recall that when f is convex, finite at x and g : X → R is convex continuous, one has (2)

∂(f + g)(x) = ∂f (x) + ∂g(x).

The following extension result is also a consequence of the Hahn-Banach theorem, as observed to us by M. Fabian. We derive it from (2). In the sequel, given a function f on X and a subspace W of X, we denote by f |W the restriction of f to W. Lemma 1. Let W be a linear subspace of a normed vector space X and ∗ ∈ let g : X → R be a convex continuous function. Given z ∈ W and zW ∗ . ∂(g|W )(z) there exists some z ∗ ∈ ∂g(z) such that z ∗ |W = zW Proof. By the Hahn-Banach theorem, there exists some y ∗ ∈ X ∗ which ∗ . Let ι extends zW W be the indicator function of W : it is defined by ιW (w) := 0 for w ∈ W, ιW (x) := +∞ for x ∈ X\W. Then, we clearly have y ∗ ∈ ∂(g + ιW )(z). Now, definition (1) shows that ∂ιW (z) = W ⊥ := {x∗ ∈ X ∗ : x∗ |W = 0} and since g is convex continuous, relation (2) ensures that ∂(g + ιW )(z) = ∂g(z) + ∂ιW (z) = ∂g(z) + W ⊥ . ∗ . Thus we can find z ∗ ∈ ∂g(z) satisfying y ∗ − z ∗ ∈ W ⊥ , hence z ∗ |W = zW

Let us recall the concept of calmness. A function f : X → R∞ finite at x ∈ X is said to be calm at x if there exist c ∈ R+ and a neighborhood V of x such that f (x) − f (x) ≥ −c kx − xk for all x ∈ V. If one can take V = X, one says that f is globally calm at x. When f is convex, the two notions coincide. The calmness rate of f at x is the infimum γf (x) of the constants c > 0 for which the preceding inequality is satisfied for some neighborhood of x. The remoteness of a nonempty subset S of a nvs is the number ρ(S) := inf{ksk : s ∈ S}. Such a simple notion enables us to give a quantitative version of a useful characterization of the subdifferentiability of a convex function g at some x, i.e. of the nonemptiness of ∂g(x). Lemma 2. A convex function g : X → R∞ finite at some x ∈ X is subdifferentiable at x iff it is globally calm at x, iff it is calm at x. Moreover, the calmness rate of g at x is equal to the remoteness ρ(∂F g(x)) := inf{kx∗ k : x∗ ∈ ∂F g(x)} of ∂F g(x). Proof. If ∂g(x) is nonempty, for any element x∗ ∈ ∂g(x) one can take c = kx∗ k to get global calmness. Conversely, if one can find c ∈ R+ such that g(x) + c kx − xk ≥ g(x) for any x ∈ X, then the sum rule yields 0 ∈ ∂g(x) + c∂ k·k (0), so that ∂g(x) is nonempty and in fact contains an element of Unauthenticated Download Date | 5/29/17 4:11 PM

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A short proof of the separable reduction theorem

cBX ∗ = c∂ k·k (0). If g is calm at x, the convex function x 7→ g(x)+c kx − xk attains a local minimum at x, hence a global minimum at x. Let us make clear some facts about convexification. Let us recall that the convex hull conv(E) of a subset E of a nvs Z is the smallest convex subset C of Z containing E. Introducing the set Nm of the m first positive integers and the canonical simplex ∆m by m n o X m Nm := {i ∈ N : 1 ≤ i ≤ m}, ∆m = (t1 , . . . , tm ) ∈ R+ , tj = 1 , j=1

the convex hull of E can be described explicitely as C := conv(E) = {x = t1 x1 + · · · + tm xm : m ≥ 1, t := (t1 , . . . , tm ) ∈ ∆m , xi ∈ E}, as is well known and easy to check. The convex hull conv(h) of a function h : Z → R∞ is the greatest convex function g majorized by h. Its epigraph is almost the convex hull of the epigraph E of h. In fact, it is the vertical closure of conv(E) : one has epis g ⊂ conv(E) ⊂ epi g, where epis g := {(x, r) ∈ X × R : r > g(x)} as easily seen. The fact that these sets are distinct has been pointed out to us by C. Zălinescu who provided the example of h : R → R given by h(0) := 1, h(x) := |x| for x ∈ R\{0}. Thus g(x) := inf

m nX

ti h(xi ) : m ≥ 1, t := (t1 , . . . , tm ) ∈ ∆m ,

i=1

o xi ∈ X, t1 x1 + · · · + tm xm = x .

Lemma 3. (a) Given a sequence (En ) of nonempty subsets of a vector space Z, the convex hull C of the union E of the En ’s is the union over p ∈ N\{0} of the convex hulls Cp of E1 ∪ · · · ∪ Ep : [ [ (3) C := conv(E) = Cp where Cp := conv En . p

n≤p

Denoting by Jm,p the set of maps j : Nm → Np , for m, p ∈ N\{0}, the set Cp is given by m o [ [ nX ti xi : t := (t1 , . . . , tm ) ∈ ∆m , xi ∈ Ej(i) . (4) Cp := m≥1j∈Jm,p

i=1

(b) Given a sequence (hn ) of functions on a vector space X, the convex hull k of the function h := inf n hn is the infimum over p ∈ N\{0} of the convex hulls kp := conv (h1 , . . . , hp ) of the functions h1 , . . . , hp . The function kp is Unauthenticated Download Date | 5/29/17 4:11 PM

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given by (5)

kp (x) := inf

inf inf

m≥1 j∈Jm,p

m nX

ti hj(i) (xi ) : (t1 , . . . , tm ) ∈ ∆m ,

i=1

xi ∈ X,

m X

o ti xi = x .

i=1

Notice that in general, the union of a family (Cp ) of convex subsets is no more convex; but when the sequence (Cp ) is increasing (with respect to inclusion), the union is convex. Similarly, the infimum of a countable family (kp ) of convex functions is convex when the sequence (kp ) is decreasing; but that is not the case if the sequence (kp ) does not satisfies this property. Proof. (a) In fact, any element of C can be written as a convex combination of a finite family of elements of E, hence is an element of Cp for some p. The reverse containment is obvious since Cp ⊂ C for all p. The right-hand side of (4) is clearly contained in Cp . Using the fact that the concatenation of an element j of Jm,p with an element j ′ of Jn,p is an element of Jm+n,p , it is easily seen that this set is convex and contains all En for n ∈ Np , it coincides with Cp . (b) Now, when En is the epigraph of a function hn , the vertical closure of Cp is the epigraph of kp := conv (h1 , . . . , hp ) , the greatest convex function majorized by h1 , . . . , hp . The right-hand side of (5) defines a function which is clearly bounded below by kp . Since it is easily seen that it is convex and bounded above by hn for all n ∈ Np , it coincides with kp . One can also derive this formula from (4) by using epigraphs. 2. A characterization of Fréchet subdifferentiability We devote the present section to a characterization of the nonemptiness of the Fréchet subdifferential ∂F f (x) of a (nonconvex, nonsmooth) function f at some point x of its domain, domf. Recall that f (x + x) − f (x) − hx∗ , xi ≥ 0}. x6=0 kxk

∂F f (x) := {x∗ ∈ X ∗ : lim inf x→0,

Clearly, f is Fréchet differentiable at x if, and only if, ∂F f (x)∩(−∂F (−f )(x)) is nonempty. Simple examples show that the nonemptiness of ∂F f (x) may frequently occur when X is a nice space; but here X is an arbitrary Banach space. We will use the following notion of approximate Fréchet subdifferential. Given an arbitrary function f : X → R and ε > 0 the ε−Fréchet subdifferential ∂Fε f (x) of f at x ∈ domf is the set of all x∗ ∈ X such that Unauthenticated Download Date | 5/29/17 4:11 PM

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there exists ρ > 0 for which f (x + x) − f (x) − hx∗ , xi ≥ −εkxk

(6)

When f is convex one has

∀x ∈ ρBX .

∂Fε f (x)

= ∂(f + εk · −xk)(x). Clearly one has \ ∂F f (x) = ∂Fε f (x). ε>0

One can give a characterization of the nonemptiness of the sets ∂Fε f (x) and ∂F f (x) which parallels the one we gave in the convex case. For such a purpose, given x ∈ X, a function f : X → R finite at x and ε, ρ > 0, we introduce the function f ε,ρ given by f ε,ρ (x) := f (x + x) − f (x) + εkxk f ε,ρ (x) := + ∞ for x ∈ X\ρB.

for x ∈ ρB,

Here B := BX is the closed unit ball of X. Lemma 4. Given an arbitrary function f : X → R∞ := R∪{+∞} and ε > 0, the ε-Fréchet subdifferential ∂Fε f (x) of f at x ∈ domf contains some element of norm at most c ∈ R+ if and only if there exists ρ > 0 such that conv(f ε,ρ )(·) ≥ −ck · k. The last relation can be rephrased more explicitely as follows: for all m ≥ 1, (t1 , . . . , tm ) ∈ ∆m , x1 , . . . , xm ∈ ρB m m

X X

ti xi . ti (f (x + xi ) + εkxi k) ≥ f (x) − c Proof. Let Then, for m

i=1 ε ∂F f (x) and let ρ > 0 be as in the definition of ≥ 1 and (t1 , . . . , tm ) ∈ ∆m , x1 , . . . , xm ∈ ρB we have f (x + xi ) + εkxi k ≥ f (x) + hx∗ , xi i i = 1, . . . , m. i=1 ∗ x ∈

this set.

Multiplying both sides by ti and summing, we obtain the expected relation by taking c := kx∗ k. Conversely, suppose conv(f ε,ρ )(·) ≥ −ck · k for some ρ > 0, c ≥ 0. Then, since conv(f ε,ρ )(0) ≤ f ε,ρ (0) = 0, we have conv(f ε,ρ )(x) ≥ −ckxk ≥ conv(f ε,ρ )(0) − ckxk, hence conv(f ε,ρ )(0) = 0 and Lemma 2 yields some x∗ ∈ ∂(conv(f ε,ρ ))(0) ∩ cBX ∗ . Then, for x ∈ ρB, we get f ε,ρ (x) ≥ conv(f ε,ρ )(x) ≥ conv(f ε,ρ )(0) + hx∗ , xi = hx∗ , xi. That ensures that x∗ ∈ ∂Fε f (x). Now let us characterize the nonemptiness of the set ∂F f (x). In the sequel (εn ) is a fixed sequence of positive numbers with limit 0. Lemma 5. Given c ∈ R+ , (εn ) → 0+ , and an arbitrary function f : X → R finite at x, one has ∂F f (x)∩cBX ∗ 6= ∅ if, and only if, there exists a sequence Unauthenticated Download Date | 5/29/17 4:11 PM

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(ρn ) of positive numbers such that, for f ε,ρ given as in the preceding lemma, one has (7)

∀x ∈ X

conv(inf f εn ,ρn )(x) ≥ −ckxk. n

x∗

∈ ∂F f (x) ∩ cBX ∗ if, and only if, there exists It could be added that a sequence (ρn ) of positive numbers such that conv(inf n f εn ,ρn )(0) = 0 and x∗ ∈ ∂F conv(inf n f εn ,ρn )(0). However, our aim is to obtain a condition which just involves an estimate about this function and no element of the dual space. Proof. Let x∗ ∈ ∂F f (x) ∩ cBX ∗ . Let ρn > 0 be such that f (x + x) − f (x) − hx∗ , xi ≥ −εn kxk

∀x ∈ ρn B

Then, for all x ∈ X and n, one has f εn ,ρn (x) ≥ and let g :=conv(inf n ∗ ∗ hx , xi hence g(x) ≥ hx , xi, in particular g(0) ≥ 0 and in fact g(0) = 0 since f εn ,ρn (0) = 0 for all n. Thus x∗ ∈ ∂g(0). Using Lemma 2, we get (7). Conversely, this last relation ensures that ∂g(0) ∩ cBX ∗ 6= ∅. Now, for every x∗ ∈ ∂g(0) we have x∗ ∈ ∂F f (x) since for each n ∈ N and each x ∈ X we have f εn ,ρn (x) ≥ g(x) ≥ hx∗ , xi, f εn ,ρn ).

hence x∗ ∈ ∂Fεn f (x) (since f εn ,ρn (0) = 0). Applying Lemmas 2 and 3 to hn := f εn ,ρn , we see that relation (7) holds if, and only if, for all p ∈ N\{0} one has (8)

∀x ∈ X

conv(h1 , . . . , hp )(x) ≥ −ckxk.

3. Simple separable reduction The following striking result has an independent interest, but it is not our final aim. Theorem 6. Let f : X → R∞ be a lower semicontinuous function finite at x and let c ∈ R+ . Then, for every separable subspace W0 of X there exists a separable subspace W containing W0 such that for any w ∈ W the relation ∂F f (w) ∩ cBX ∗ 6= ∅ holds if, and only if, ∂F (f |W )(w) ∩ cBW ∗ 6= ∅ holds. Proof. Without loss of generality, we may assume that εn ≤ 1 for all n and, changing f into c−1 f if necessary, c ≤ 1. We construct a sequence ((Wk , Ak ))k of pairs such that Wk is a separable subspace of X and Ak is a dense countable subset of Wk . We start with W0 and take for A0 any dense countable subset of W0 . Assume we have constructed (Wn , An ) for n = 0, . . . , k. Since f is lsc, for all x ∈ X we have η(x) := sup{r ∈ R+ : inf f (B(x, r)) > −∞} > 0. Unauthenticated Download Date | 5/29/17 4:11 PM

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Given a ∈ X, k, m, p ∈ N\{0}, q > 0, ρ := (ρ1 , . . . , ρp ) ∈ (0, η(a)/2)p , j ∈ Jm,p = (Np )m (so that j includes the data of m and p), t := (t1 , . . . , tm ) ∈ ∆m , let X(a, ρ, j, q, t) := {(x1 , . . . , xm ) ∈ X m : xi ∈ 2ρj(i) B,

(9)

kt1 x1 + · · · + tm xm k < q} and let (w1 , . . . , wm ) := w(a, ρ, j, k, q, t) ∈ X(a, ρ, j, q, t) be such that inf

m nX

ti (f (a + xi ) + εj(i) kxi k) : (x1 , . . . , xm ) ∈ X(a, ρ, j, q, t)

i=1

≥

m X

o

ti (f (a + wi ) + εj(i) kwi k) − 1/k.

i=1

Let Wk+1 be the closed linear subspace generated by Wk and the vectors wi (a, ρ, j, k, q, t) for a ∈ Ak , m, p ∈ N\{0}, i ∈ Nm , j ∈ Jm,p := {j : Nm → Np } = (Np )m , q ∈ Q, ρ ∈ (0, η(a)/2)m ∩ Qm , t := (t1 , . . . , tm ) ∈ ∆m ∩ Qm . Clearly Wk+1 is separable, so that we can take a dense countable subset Ak+1 of Wk+1 containing Ak . Let W be the closure of the union of the Wk ’s. Let w ∈ W be such that ∂F (f |W )(w) ∩ cBW ∗ 6= ∅. Thus, there exists some decreasing sequence (ρn )n of positive rational numbers in (0, η(w)/4) ∩ (0, 1) such that, setting W fn,w (v) := f (w + v) − f (w) + εn kvk + ι3ρn B (v),

v∈W

fn,w (x) := f (w + x) − f (w) + εn kxk + ιρn B (x),

x∈X

where ιrB denotes the indicator function of the ball rB, one has (10)

∀p ≥ 1, ∀v ∈ W

W W conv(f1,w , . . . , fp,w )(v) ≥ −ckvk.

Let us prove that, for all α > 0, we have (11)

∀p ≥ 1, ∀x ∈ X

conv(f1,w , . . . , fp,w )(x) ≥ −ckxk − α.

This relation means that for all m, p ∈ N\{0}, all j : Nm → Np , all t := (t1 , . . . , tm ) ∈ ∆m and all x1 , . . . , xm ∈ X with x := t1 x1 + · · · + tm xm , we have (12)

t1 fj(1),w (x1 ) + · · · + tm fj(m),w (xm ) ≥ −ckxk − α.

We may suppose that t1 , . . . , tm are rational numbers. It also suffices to consider the case in which xi ∈ ρj(i) B for all i ∈ Nm . Let q be a rational number satisfying kxk < q < kxk + α/5, and let β ∈ (0, α/5) be such that β < η(w)/2, kxk + β < q. We take k ≥ β −1 with d(w, Wk ) < min(β, ρp ) and a ∈ Ak such that ka − wk ≤ min(β, ρp ). Then B(a, η(w)/2) ⊂ B(w, η(w)) so that η(a) ≥ η(w)/2. For i ∈ Nm , setting vi := xi + w − a, we note that we have kxi − vi k ≤ β, kvi k ≤ ρj(i) + ρp ≤ 2ρj(i) , ρi < η(w)/4 ≤ η(a)/2 and Unauthenticated Download Date | 5/29/17 4:11 PM

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kt1 v1 + · · · + tm vm k ≤ kt1 x1 + · · · + tm xm k + max kvi − xi k ≤ kxk + β < q. i

Thus vi ∈ X(a, ρ, j, q, t). The relation w + xi = a + vi and the triangular inequality justify the first of the following string of inequalities, while the second one stems from the definition of wi ; the relations a + wi = w + zi , for zi := a − w + wi , explain the next ones: m X

ti (f (w + xi ) + εj(i) kxi k)

i=1

≥

m X

ti (f (a + vi ) + εj(i) kvi k − εj(i) kxi − vi k)

≥

m X

ti (f (a + wi ) + εj(i) kwi k) − 1/k − β

≥

m X

ti (f (w + zi ) + εj(i) ka − w + wi k − εj(i) ka − wk) − 2β

i=1

i=1

≥

i=1 m X

ti (f (w + zi ) + εj(i) kzi k) − 3β.

i=1

Since zi := a − w + wi ∈ W ∩ 3ρj(i) B, for z := t1 z1 + · · · + tm zm , from (10), we get m m X X ti fj(i),w (zi ) ≥ ti f (w + zi ) − f (w) + εj(i) kzi k − 3β i=1

i=1

=

m X

W ti fj(i),w (zi ) − 3β

i=1

≥ −c kzk − 3β. Now, since zi − wi = a − w and t1 + · · · + tm = 1, by (9) we have the estimate kzk ≤ kt1 wi + · · · + tm wm k + ka − wk ≤ q + β ≤ kxk + α/5 + β. Since c ≤ 1 and β ≤ α/5, relation (12) is established.

4. Application to fuzzy calculus Using the previous results, one can prove the following crucial separable reduction property for sums. Theorem 7. Let W0 be a separable subspace of a Banach space X, and let f : X → R∞ be a lsc function and g : X → R be a convex continuous function. Then, there exists a separable subspace W of X containing W0 Unauthenticated Download Date | 5/29/17 4:11 PM


661

such that for any w, z ∈ W the relation (∂F f (w) + ∂g(z)) ∩ cBX ∗ 6= ∅ holds if, and only if, (∂F (f |W )(w) + ∂(g|W )(z)) ∩ cBW ∗ 6= ∅ holds. Proof. Let us first consider the case g is a continuous linear form. Then, since ∂F (f + g)(w) = ∂F f (w) + g, with a similar relation for the restrictions to W, the result follows from Theorem 6 applied to f + g. An examination of its proof shows that the construction of W for f remains valid for f + g as the same sequence (ρn ) can be used for f + g and W (f + g)W n,w = fn,w + g|W , (f + g)n,w = fn,w + g,

conv(f1,w + g, . . . , fp,w + g) = conv(f1,w , . . . , fp,w ) + g W + g| , . . . , f W + g| ). and a similar relation with conv(f1,w W W p,w Now let us consider the general case. Given f and a separable subspace W0 of X, let W be a separable subspace containing W0 associated with f. ∗ ∈ ∂ (f | )(w), z ∗ ∈ ∂(g| )(z) be such that Let w, z ∈ W and let wW F W W W ∗ ∗ kwW + zW k ≤ c. Using Lemma 1, we can find some z ∗ ∈ ∂g(z) such that ∗ . Then, by the preceding special case, we can find w ∗ ∈ ∂f (w) z ∗ |W = zW such that kw∗ + z ∗ k ≤ c, i.e. (w∗ + z ∗ ) ∈ (∂F f (w) + z ∗ ) ∩ cBX ∗ ⊂ (∂F f (w) + ∂g(z)) ∩ cBX ∗ .

In the next statement, following [10], [11] and subsequent papers, we say that X is a subdifferentiability space for the Fréchet subdifferential ∂F if for any lower semicontinuous (lsc) function on X the set G of points (w, f (w)) such that ∂F f (w) is nonempty is dense in the graph of f. We say that X is a reliable space for the Fréchet subdifferential ∂F if whenever f : X → R∞ is a lsc function, g : X → R is a convex continuous function and f + g attains its minimum on X at x ∈ X, for every ε > 0 there exist w, z ∈ B(x, ε) with |f (w) − f (x)| ≤ ε such that the relation (∂F f (w)+∂g(z))∩εBX ∗ 6= ∅ holds. The space X is said to be a trustworthy space for the Fréchet subdifferential ∂F if the same holds when g is a Lipschitzian function and ∂g(z) is replaced by ∂F g(z). Theorem 8. Any Asplund space X is a Fréchet subdifferentiability space. In fact, for a Banach space X the following properties are equivalent: (a) (b) (c) (d)

X X X X

is is is is

an Asplund space; a reliable space for the Fréchet subdifferential ∂F ; a trustworthy space for the Fréchet subdifferential ∂F ; a subdifferentiability space for the Fréchet subdifferential ∂F .

Proof. (a)⇒(b) Let ε > 0, a lsc function f : X → R∞ and a convex continuous function g : X → R be given such that f + g attains its minimum on X at x ∈ X. Let W0 := Rx and let W be a separable subspace of X containing W0 associated to f as in Theorem 7. Since X is Asplund, the Unauthenticated Download Date | 5/29/17 4:11 PM

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J.-P. Penot

separable subspace W has a separable dual, hence a Fréchet smooth bump function. A classical penalization procedure then shows that there exist w, z ∈ B(x, ε) such that |f (w) − f (x)| ≤ ε and (∂F (f |W )(w) + ∂(g|W )(z)) ∩ εBW ∗ 6= ∅. Then, we conclude from Theorem 7 that (∂F f (w) + ∂g(z)) ∩ εBX ∗ 6= ∅. (b)⇒(c) This is a general implication, valid for any subdifferential; see [13], [15]. (c)⇒(d) Again, this is a general (and easy) fact. (d)⇒(a) Let f : X → R be a continuous convex function and let x ∈ X, ε > 0 be given. Let g := −f. Since X is a subdifferentiability space for ∂F there exists some x ∈ B(x, ε) such that ∂F g(x) is nonempty. Then f is Fréchet differentiable at x. Thus f is densely Fréchet differentiable and X is an Asplund space. Acknowledgements. The author is grateful to A.D. Ioffe for giving him incentives to devise a simple proof and to give him access to notes used in the last section, to M. Fabian for the suggestion of putting apart the argument of Lemma 1 and to C. Zălinescu for suggesting some corrections.

References [1] J. M. Borwein, Q. J. Zhu, Techniques of Variational Analysis, CMS Books in Maths 20, Springer (2005). [2] I. Ekeland, G. Lebourg, Generic Fréchet differentiability and perturbed optimization problems in Banach spaces, Trans. Amer. Math. Soc. 224 (1976), 193–216. [3] M. Fabian, Subdifferentials, local ε-supports and Asplund spaces, J. London Math. Soc. 34 (1986), 568–576. [4] M. Fabian, On classes of subdifferentiability spaces of Ioffe, Nonlinear Anal. 12 (1988), 568–576. [5] M. Fabian, Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolin. Math. Phys. 30 (1989), no 2, 51–56. [6] M. Fabian, B. S. Mordukhovich, Smooth variational principles and characterizations of Asplund spaces, Set-Valued Anal. 6 (1998), 381–406. [7] M. Fabian, B. S. Mordukhovich, Separable reduction and supporting properties of Fréchet-like normals in Banach spaces, Canadian J. Math. 51 (1999), 26–48. [8] M. Fabian, B. S. Mordukhovich, Separable reduction and extremal principles in variational analysis, Nonlinear Anal. 49 (2002), 265–292. [9] M. Fabian, N. V. Zhivkov, A characterization of Asplund spaces with the help of local ε-supports of Ekeland and Lebourg, C. R. Acad. Bulgare Sci. 38 (1985), no 6, 687–674. [10] A. D. Ioffe, On subdifferentiability spaces, Ann. New York Acad. Sci. 410 (1983), 107–119. [11] A. D. Ioffe, Subdifferentiability spaces and nonsmooth analysis, Bull. Amer. Math. Soc. 10 (1984), 87–90. Unauthenticated Download Date | 5/29/17 4:11 PM


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[12] A. D. Ioffe, Separable reduction theorem for approximate subdifferentials, C. R. Acad. Sci. Paris, Sér. I Math. 323 (1996), No. 1, 107–112. [13] A. D. Ioffe, Fuzzy principles and characterization of trustworthiness, Set-Valued Anal. 6 (1998), 265–276. [14] B. Mordukhovich, Variational Analysis and Generalized Differentiation I, Basic Theory, Springer, Berlin (2006). [15] Q. J. Zhu, The equivalence of several basic theorems for subdifferentials, Set-Valued Anal. 6 (1998), 171–185. ´ ´ LABORATOIRE DE MATHEMATIQUES APPLIQUEES CNRS UMR 5142 ´ DES SCIENCES FACULTE ´ DE PAU ET DES PAYS DE L’ADOUR UNIVERSITE B.P. 1155, 64013 PAU CEDEX, FRANCE E-mail: [email protected]

Received July 13, 2009.

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