On Fr´ echet differentiability of convex functions on Banach spaces
Wee-Kee Tang*
Abstract. Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz functionfdefined on a separable Banach space are studied.
The conditions are in terms of a majorization offby aC1-smooth function, separability of the boundary forf or an approximation off by Fr´echet smooth convex functions.
Keywords: Fr´echet differentiability, convex functions, variational principles, Asplund spaces
Classification: 46B03
It is known that on a given separable Banach spaceX all continuous convex functions are generically Fr´echet differentiable if and only ifX∗ is separable, and if and only ifXadmits aC1-smooth bump function. In this case, every equivalent norm inX can be uniformly approximated by Fr´echet smooth equivalent norms on bounded sets.
The purpose of this note is to generalize these results. We give some equivalent conditions for the generic Fr´echet differentiability of a given Lipschitz convex function defined on a separable Banach space in terms of the properties of the functionf rather than that ofX. In this setting, we cover some continuous convex functions defined on separable non-Asplund spaces. For instance ifk · k denotes the Hilbertian norm onl2 andT is a continuous linear map of a separable Banach space X into l2, then any Lipschitz convex function f defined on X such that f(x)≤ kT(x)k2 forx∈X satisfies the assumptions in Theorem 1. At the end of this note, we show how the methods from variational principles can be applied to find a sufficient condition for thew∗-lower semicontinuity of convex functions.
A standard notation is used in this paper. We denote by ∂f(x) the sub- differential of a continuous convex function f at x ∈ X (cf. [DGZ], [Ph]), and
∂f(X) = S
x∈X∂f(x). If f is defined on X, f∗ denotes the Fenchel dual (or conjugate) off, i.e.f∗(x∗) = sup{(x∗, x)−f(x) :x∈X}, forx∗∈X∗. A convex continuous function is said to be generically Fr´echet differentiable if it is Fr´echet differentiable on a denseGδ set. A subsetB⊂∂f(X) is called a boundary forf ifB intersects∂f(x) for eachx∈X (see e.g. [G]). By a selector for∂f we mean a single-valued mapping s: X →X∗ such that s(x) ∈∂f(x) for every x ∈X.
* Supported by NSERC (Canada).
Unless stated otherwise, all topological terms in dual Banach spaces refer to the norm topology of these spaces. We refer to [Ph] and [DGZ] for some unexplained notions and results used in this note.
A main result in this note is the following statement.
Theorem 1. LetX be a separable Banach space and f be a Lipschitz convex function defined onX. The following are equivalent.
(i) The set∂f(X)is separable.
(ii) There is a selectorsfor∂f such thats(X) ={s(x) :x∈X}is separable.
(iii) There is a continuously Fr´echet differentiable functionφsuch thatφ≥f onX.
(iv) f can be approximated uniformly onX by Fr´echet differentiable convex functions.
(v) If his a convex function on X such thath≤f onX, thenhis generically Fr´echet differentiable onX.
Proof: Clearly (i)⇒ (ii). We shall show (ii)⇒(i) using Simons’ lemma ([S]).
Put B = s(X) and let γ = inf{f∗(y∗) : y∗ ∈ B} < ∞. We show that C :=
domf∗⊂convB. If this is not so, pickyo∗∈C\convB. By separation theorem, there exist z ∈ X∗∗ and α, β ∈ R such that z(y∗o) > β > α > z(y∗) for each y∗ ∈B. Without any loss of generality, we assume that β−α2 > f∗(y∗o)−γ. For everyx∈X, define a functionhx∈l∞(B) by
hx(x∗) = (x∗, x)−f∗(x∗).
Let E = {x ∈ X : kxk ≤ kzk, x(y∗o) > β}. Since B is separable, there exists a sequence {xn} in E such that xn converges to z in the topology of pointwise convergence on B. Define a sequence hn ∈ l∞(C) by hn(x∗) = hxn(x∗) for x∗ ∈C. Note that for anyx= P∞
k=1
λkxk, whereλk≥0 and P∞
k=1
λk = 1, we have s(x)∈B and
Xλkhk(s(x)) =hx(s(x)) =f(x) = sup{(x∗, x)−f∗(x∗) :x∗ ∈C}
= sup{(x∗,X
λkxk)−f∗(x∗) :x∗ ∈C}= sup{X
λkhk(x∗) :x∗∈C}.
Sincez(y∗)< αfor ally∗ ∈B,we have lim supxn(y∗)≤αfor ally∗ ∈B. Thus lim suphn(y∗)≤α−f∗(y∗) for ally∗∈B and consequently sup{lim suphn(y∗) : y∗∈B} ≤α−γ. By Simons’ lemma there isg∈conv{hn},g= PN
k=1
̺khk,̺k≥0, PN
k=1
̺k= 1 such that
sup{g(x∗) :x∗ ∈C} ≤ α+β 2 −γ.
On the other hand,g(yo∗) = PN
k=1
̺khk(yo∗) = (y∗o, PN
1
̺kxk)−f∗(yo∗)> β−f∗(y∗o) and thusβ−f∗(yo∗) < α+β2 −γ. Therefore β−α2 < f∗(yo∗)−γ. This contradiction shows that (ii) implies (i).
(iii)⇒(i). We follow the idea in [F]. Letx∈X,q∈∂f(x) andǫ >0 be given.
The functionφ−qis a bounded below continuous function on X. By Ekeland’s variational principle, there is axq such that for eachh∈X andt >0,
(φ−q)(xq+th)≥(φ−q)(xq)−ǫkhkt.
Hence,
kφ′(xq)−qk∗≤ǫ.
Therefore∂f(X)⊂ {φ′(x) :x∈X}. Sinceφ′ is continuous and X is separable, the set{φ′(x) :x∈X}and thus also∂f(X) are separable.
(i)⇒(v). By using the above argument for the functionshandf, we see that
∂h(X) ⊂∂f(X). Therefore ∂h(X) is also separable and the statement follows immediately from the proof of Theorem 1 in [Pr-Z] (see also [Ph, Theorem 2.11]).
(v)⇒(i). Since ∂f(X)⊂domf∗, it suffices to show that domf∗ is separable.
We split domf∗intow∗-compact setsCnand show that allCnare norm separable.
We putCn={x∗∈X∗:f∗(x∗)≤n}, and note that domf∗=∪∞n=1Cn.
Assume for somen∈N, the setCnis not norm separable. SinceCnis compact and metrizable in thew∗-topology, we find aw∗-compact subsetA⊂CN andǫ >
0 such that everyw∗-slice has diameter greater thanǫ >0 (see the proof of [Ph, Theorem 2.19]). Defineh(x) = suphA, xi −N, x∈X. Then h(x)≤sup{hx∗, xi − f∗(x∗) :x∗ ∈CN}, and the function his nowhere Fr´echet differentiable (see the proof of [Ph, Lemma 2.18]). Ash≤f onX, we obtain a contradiction.
(i)⇒ (iv). LetY =span{∂f(x) :x∈X}. Since Y is norm separable, there is an equivalent normk · k on X such that its dual normk · k∗ onX∗ is locally uniformly rotund at points of domf∗. In other words, if y ∈domf∗, yk ∈X∗, and lim(kykk∗22+kyk∗2 − kyk2+yk∗2) = 0, then limkyk−yk∗= 0 (see e.g. the proof of [DGZ, Proposition IV.5.2]).
Now, define a sequence of functions{hn}onX∗byhn(x∗)=f∗(x∗)+4n14kx∗k∗2. Clearly domhn= domf∗. Note that ifn∈Nand lim(hn(y)+h2 n(yk)−hn(y+y2k)) = 0, then limkyk−yk = 0. Definegn:=fn4k · k2, the infimal convolution of f andn4k · k2. Note that the functiongnis a convex continuous function onX for allnandgn∗ =hn.
Givenn∈N, x∈X andy∈∂gn(x), note thathnis rotund aty with respect toxin the sense of [ As-R], i.e., for everyǫ >0, there existsδ >0 such that
{v:hn(y+v)−hn(y)−(x, v)≤δ} ⊂ǫBX∗.
Indeed, if this is not so, there exists anǫ >0 such that for all k∈N, there is avk, withkvkk> ǫand
1
2hn(y+vk)−1
2hn(y)−(x,vk 2 )≤ 1
2k. Sincehn=gn∗, we havex∈∂hn(y), and thus
(x,vk
2 )≤hn(y+vk
2 )−hn(y)
Putting these two inequalities together, we obtain for everyk∈N, hn(y) +hn(y+vk)
2 −hn(y+vk 2 )≤ 1
2k.
From the local uniform convexity ofhn, we have limkvkk= 0, a contradiction.
By [As-R, Proposition 4], gn is Fr´echet differentiable atxwith the derivativey.
By the proof of Lemma 2.4 in [MPVZ], one can show that limgn=f uniformly onX.
(iv) ⇒ (iii). By (iv), there exists a Fr´echet differentiable convex function ψ such that|ψ(x)−f(x)| ≤ 12 for everyx∈X. Thenψ+ 1 is a desired function.
This completes the proof of Theorem 1.
Note that in Theorem 1, the implications (iii)⇒(i)⇒(v) are still valid with- out requiringf to be Lipschitz. The assumption of separability ofX in the state- ment of Theorem 1 cannot be dropped in general. Indeed, Haydon constructed a nonseparable space X where all convex continuous functions are generically Fr´echet differentiable and yet no equivalent norm can be approximated uniformly on bounded sets by Fr´echet differentiable convex functions (see e.g. [DGZ]).
Note also that in Theorem 1, it is crucial that the function f be defined on the whole ofX, as there may exist nowhere differentiable norms bounded on the open ball by constant functions.
The following statement shows how Ekeland’s variational principle can be used in questions onw∗-lower semicontinuity of convex functions.
Theorem 2. Let X be a Banach space and f be a w∗-lower semicontinuous Fr´echet differentiable function on X∗. Then every norm-lower semicontinuous convex function g on X∗ such that g ≤ f on X∗ is w∗-lower semicontinuous onX∗.
Proof: We first note thatf′(X∗) :=∪{f′(y) :y∈X∗} ⊂X. Indeed, for anyy∈ X∗,f′(y) isw∗-lower semicontinuous onBX∗, as it is a uniform limit ofw∗-lower semicontinuous functions on BX∗. Since f′(y) is linear, f′(y) is w∗-continuous onBX∗. By Banach-Dieudonn´e Theorem,f′(y) is w∗-continuous onX∗. Hence f′(y)∈X for anyy∈X∗.
We claim that domg∗ ⊂X. Indeed, for anyh∈domg∗, we have sup{h(x∗)− g(x∗) :x∗∈X∗}<∞. This implies thatf−his bounded below. As in the proof of (iii)⇒(i), we can showh∈f′(X∗). Thereforeh∈X.
Since g is norm-lower semicontinuous, we have g =g∗∗↾X∗. However, g∗∗ = (g∗)∗ = (g∗↾domg∗)∗ = (g∗↾X)∗. Hence g is a dual to a function defined on X,
thereforegisw∗-lower semicontinuous.
Acknowledgements. The author would like to thank Professor V. Zizler for his guidance and Professor M. Fabian for his helpful suggestions and enlightening discussions on this paper.
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Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
(Received August 2, 1994)
