# View of On $$\beta$$-differentiability of norms

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Rev. Anal. Num´er. Th´eor. Approx., vol. 33 (2004) no. 2, pp. 141–148 ictp.acad.ro/jnaat

ON β-DIFFERENTIABILITY OF NORMS

VALERIU ANISIU

Dedicated to Professor Elena Popoviciu on the occasion of her 80th birthday.

Abstract. In this note we give some characterizations for the differentiability with respect to a bornology of a continuous convex function. The special case of seminorms is treated. A characterization of this type of differentiability in terms of the subgradient of the function is also obtained.

MSC 2000. 58C20, 46A17, 46G05.

Keywords. Convex function, differentiability, bornology, subgradient.

RESULTS

Let E and E1 be Banach spaces, U an open subset in E and xU. A functionf :UE1 is said to be Gˆateaux differentiable atx if there exists a linear continuous mapping denoted df(x) :EE1 such that for eachh inE one has

(1) df(x)(h) = lim

t→0+

1

t(f(x+th)f(x)).

The functionf is said to beFr´echet differentiableatxif there exists a linear continuous mapping denoted f0(x) : EE1 such that for each ε > 0, there existsδ >0 satisfying

(2) kf(x+h)f(x)−f0(x)(h)k ≤εkhk, for each hBE(x, δ).

The two linear mappings df(x), f0(x) are the Gˆateaux and Fr´echet differ- entials and are unique (when they exist).

In the sequel, we shall be interested only by real functions (i.e. E1 =R).

When a real function f is also convex on an open convex set UE, then the limit in (1) exists and is denoted by d+f(x); this directional derivative is generally only sublinear and the Gˆateaux differentiability off atxis equivalent with the linearity of d+f(x), or with the fact that

d+f(x)(h) =−d+f(x)(−h) [=: df(x)(h)], for each hE.

This research was supported in part by CNCSIS under Contract no. 46474/97 code 14.

“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, 1 Kog˘alni- ceanu St., 400084 Cluj-Napoca, Romania, e-mail: [email protected]

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It is obvious that any Fr´echet differentiable function is also Gˆateaux dif- ferentiable, and the two differentials coincide. The converse is not true, even for convex functions. For example, the norm of the Banach space1 is known to be nowhere Fr´echet differentiable, but it is Gˆateaux differentiable at those points (xn)n∈N having only nonzero components.

It is well known that the functionf is Fr´echet differentiable atxif and only if it is Gˆateaux differentiable at x and the limit (1) is uniform with respect to hB[0,1] (=the closed unit ball in E) or, equivalently, with respect to any bounded subset of E. This remark allows a useful generalization of the differentiability.

Letβ be a nonempty family of bounded sets inE whose union is E,which is directed with respect to ⊆ (i.e., for each B1, B2β there exists B3β such that B1, B2B3) and is invariant under scalar multiplication. Such a family is namedbornology in Phelps’ monograph [3].

The function f is said to beβ-differentiable at the point x iff is Gˆateaux differentiable at x and the limit (1) is uniform in hB for each Bβ.

This turns out to be equivalent with the convergence in the uniform struc- ture Fβ(E,R). We shall denote by τβ the topology induced by this uniform structure.

The following interesting special cases of a bornology arise (as pointed out in [3]):

β=G= the family of all finite subsets inE (generating the Gˆateaux differentiability);

β=F = the family of all bounded subsets in E (generating the Fr´echet differentiability);

β = H = the family of all compact subsets in E (generating the Hadamard differentiability);

β=W = the family of all weak compact subsets inE (generating the strong Hadamard differentiability).

One obviously has the inclusions: GβF, GHWF; if f is β2-differentiable and β1β2, then f is also β1-differentiable and the two differentials coincide.

Theorem 1. Let f be a continuous convex function on an open convex subset U in the normed space E and β a bornology on E. Then f is β- differentiable at xU if and only if, for each Bβ, the limit

(3) lim

t→0+

1

t(f(x+th) +f(x−th)−2f(x)) = 0, holds uniformly for hB.

Proof. Necessity. Let B be an arbitrary subset in β. Using (1) for B and

−B one obtains the equalities df(x)(h) = limt→0+1

t(f(x+th)f(x)) and df(x)(−h) = limt→0+1t(f(x−th)f(x)), which hold uniformly for hB;

by addition the desired conclusion follows.

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Sufficiency. Choose Bβ, ε > 0. Using the continuity of f, one can select a subgradient x∂f(x). The hypothesis guarantees the existence of a positive number δ such that f(x+th) +f(xth)−2f(x) < tε, for each hB and t ∈(0, δ). (B is bounded, so, for sufficiently small δ >0 one has x±thB.)

We have

hx, thi ≤f(x+th)f(x), hx,−thi ≤f(x−th)f(x), and for 0< t < δ,hB one obtains:

0≤f(x+th)f(x)− hx, thi

= f(x+th) +f(xth)−2f(x)+ f(x)−f(x−th)− hx, thi

≤εt+ 0 =εt,

which implies that (1) holds uniformly for hB.

Remark1. a)For the Fr´echet differentiability it is sufficient that the limit (3) holds uniformly on the unit sphere SE; for the Gˆateaux differentiability, the pointwise limit in (3) suffices.

b) The continuity condition imposed on the convex function f cannot be omitted when E is infinite dimensional; in fact it is sufficient to consider a linear discontinuous functional f (cf. [5, p. 251]); in this case, df(x) = f is

not continuous.

Corollary2. Letfn(n∈N)be a sequence of continuous convex functions on an open convex subset in a Banach space E endowed with a bornology β.

If the seriesPn∈

N fn is pointwise convergent having a continuous sumf, and f isβ-differentiable at a pointx0, then each function fn is β-differentiable at x0.

Proof. The statement follows immediately from the preceding theorem, using the relations:

0≤ X

n∈N 1

t(fn(x+th) +fn(x−th)−2fn(x))

= 1t(f(x+th) +f(xth)−2f(x))

(valid forxU,t >0,hBβ provided that x+thU).

The (semi)norms are important special cases of convex functions. The next result represents a simple characterizations for theβ-differentiability of a norm, extending a theorem of Smulian [1].

Theorem 3. Let E be a normed space endowed with a bornology β and x a point on the unit sphere SE of E.

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The norm k·k isβ-differentiable atx if and only if the following condition holds: for all sequences xn, ynSE satisfying xn(x) → 1, yn(x) → 1 one has xnyn →0 in Fβ(E,R).

Proof. Necessity. Let Bβ, ε > 0, xn, ynSE satisfy xn(x) → 1, yn(x) → 1. Choosing B0β such that B ∪(−B) ⊆ B0 and applying the preceding theorem, there existsδ >0 such that fort∈(0, δ] one has

kx+thk+kx−thk<2 +εt≤2 +εδ, for each hB0.

The hypothesis implies the existence of a positive integer n0 such that for nn0:

|1−xn(x)|+|1−yn(x)|< εδ.

We have

xn(x+th) +yn(x−th)≤ kx+thk+kx−thk ≤2 +εδ, hence

xn(th)−yn(th)≤1−xn(x) + 1−yn(x) +εδ <2εδ, fornn0. By takingt=δ, one obtainsxn(h)−yn(h)<2ε, fornn0,hB0.

ForhB, we have ±h∈B0, and the last inequality implies

|xn(h)−yn(h)|<2ε, for each nn0, hB.

Sufficiency. Suppose by contradiction that k·kis not β-differentiable atx, and hence there exists ε > 0, Bβ,hnB\{0},tn >0 such that tn → 0, kx+tnhnk+kx−tnhnk ≥2 +εtn.

Choosing xn, ynSE such that xn(x+tnhn) ≥ kx+tnhnk − ktnhnk/n and yn(x−tnhn)≥ kx−tnhnk − ktnhnk/n one obtains

1≥xn(x)

=xn(x+tnhn)−xn(tnhn)

≥ kx+tnhnk −ktnnhnk − ktnhnk

≥1−ktnnhnk −2ktnhnk, hence xn(x)→1.

Similarly, yn(x)→1.

Because

xn(x+tnhn) +yn(x−tnhn)≥2 +εtn−2ktnnhnk, we have

xn(hn)−yn(hn)≥ε−2khnnk2ε, fornn0,

in contradiction withxnyn→0 inFβ(E,R)

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Iff is a continuous convex function on an open convex subsetU of a Banach space E, then the subdifferential ∂f is a set-valued operator, having convex, nonempty weak compact values in E.

We shall obtain a characterization of the β-differentiability for f in terms of the subdifferential operator ∂f. Such characterizations are known for the Fr´echet and Gˆateaux differentiability, and are very useful in the analysis of the smoothness off; in the same time such results motivated an intensive research on the set-valued operators.

If (X, τ1 ), (Y, τ2 ) are topological spaces, a set-valued operatorT :X →2Y is said to be τ12 upper semicontinuous (u.s.c.) atxX, if for each subset Wτ2 containing T(x),there exists Vτ1 containing x such that T(V) =

∪{T(v) :vV} ⊆W.

The set dom(T) :={x∈X:T(x)6=∅} is thedomain of T.

We are interested in the case when the operatorT acts betweenE and 2E, where E is a Banach space. Denoting by k·k the norm in E and by k · k its dual norm in E we shall consider the strong topology τk·k (generated by the norm) on E, and the topology τβ of the β-convergence on E, where β is a bornology on E. We remind that τF = τk·k, where F is the Fr´echet bornology (of all bounded subsets), andτGis the weak topology (Gdenoting the Gˆateaux bornology).

Proposition 4. Let E be a normed space, β a bornology on E, x a point in E and T :E → 2E a set-valued operator. Then, the following statements are equivalent:

(i) T is τk·kβ upper semicontinuous at x.

(ii) For each Wτβ, T(x)⊆W, (xn)n∈NE with kxnxk →0, there existsn0 ∈N, such that for nn0, T(xn)⊆W.

(iii) For each Wτβ, T(x) ⊆ W, there exists δ > 0, such that for r ∈ (0, δ], T(B[x, r])⊆W.

If, furthermore, T(x) is a singleton {x0}, the above conditions are also equivalent with

(iv) For each (xn)n∈NE, withkxnxk →0 it follows that

(4) lim

n→∞sup| hxx0, hi |:xT(xn), h∈B = 0, (B ∈β) and, for β=F, (4) may be reformulated as

(5) lim

r→0+diamT(B[x, r]) = 0.

Proof. The proof is standard, similar to Heine’s theorem in general topology.

We shall need the following result (see [3], [2]):

Theorem 5. Let E be a normed space, f a convex continuous function defined on an open convex set DE. Then the subdifferential ∂f :D→2E is a τk·kG upper semicontinuous operator.

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Note that in the general case, ∂f will not be τk·kβ upper-semicontinuous for an arbitrary bornology β as the following example shows:

Example 1. Let E = 1 be the Banach space of all summable sequences endowed with the norm kxk=Pn∈

N|x(n)|, and f :E →R, f(x) =kxk. For hE, we have

d+f(x)(h) = lim

t→0+

X

n∈N

|x(n)+th(n)|−|x(n)|

t

=X

n∈N

limt→0

|x(n)+th(n)|−|x(n)|

t

= X

n∈N,x(n)6=0

(signx(n))h(n) + X

n∈N,x(n)=0

|h(n)|

(the permutation of the limit and sum symbols can be legitimated by using the Weiersrass theorem, or the dominated convergence theorem from mea- sure theory applied to the sum as a discrete integral). The function f is G- differentiable at x if and only if d+f(x)(h) =−d+f(x)(−h), (h ∈ E), which means:

X

n∈N,x(n)=0

|h(n)|= 0, for each hE,i.e., x(n)6= 0,(n∈N).

Choose now x`1, x(n) =αn>0 (n∈N);then f is G-differentiable at x.

Defining xp = (α1, α2, ..., αp,0,0...), i.e. xp(n) =αn fornp and xp(n) = 0 for n > p, we obviously have kxpxk →0.

But d+f(xp)(h) = h(1) +...+h(p) +|h(p+ 1)|+|h(p+ 2)|+..., and by taking xp(h) =h(1) +...+h(p), one obtains:

xpE, xp ≤d+f(xp), hence xp∂f(xp).

On the other hand,kdf(x)−xpk = 1, so ∂f is not τk·kF u.s.c. at x (cf.

(iii), with W =B(df(x),1)).

In this example,f is notF-differentiable atx. This fact will follow from the next theorem which contains also a refinement of the preceding proposition

Theorem6. LetE be a normed space,βa bornology,f a continuous convex function on an open convex set DE, which is β-differentiable at xD.

Then the subdifferential ∂f : D → 2E is an τk·kβ upper semicontinuous operator.

Proof. Suppose by contradiction that∂f is notτk·kβ u.s.c. atx. Applying (iv) from Proposition 4 one obtains that there existxnE, withkxn−xk →0, ε >0,Bβ,hnB,xn∂f(xn), such that

| hxnx0, hni |>2ε, (n∈N), wherex0= df(x).

ChoseB0β such that B∪(−B)⊆B0.

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Interchanging if necessary hn with−hn(∈B0), we will have (6) hxnx0, hni>2ε.

From theβ-differentiability off atx, there existsδ >0, such thatB[x, δm]⊆ D, wherem >0 is chosen such thatB0B[0, m], and

f(x+th)f(x)− hx, thi ≤tε, (t∈(0, δ], h∈B0).

Hence

(7) f(x+thn)−f(x)− hx, thni ≤tε, (n∈N, t∈(0, δ]).

Using the fact that xn∂f(xn), one obtains hxn, x+δhnxni ≤ f(x+ δhn)−f(xn),hence

(8) hxn, δhni ≤f(x+δhn)−f(x) +hxn, xnxni+f(x)f(xn).

From (6), (7) and (8) we have 2εδ <hxnx0, δhni

=hxn, δhni − hx0, δhni

≤f(x+δhn)−f(x) +hxn, xnxni+f(x)−f(xn)− hx0, δhni

=(f(x+δhn)−f(x)− hx0, δhni) +hxn, xnxni+f(x)−f(xn)

≤εδ+kxnkkxnxnk+|f(x)−f(xn)|.

The convex function f being continuous, it is locally Lipschitz, hence the sequencekxnkis bounded (by the Lipschitz constant). Forn→ ∞one obtains

2εδ≤εδ, a contradiction.

Theorem7. LetE be a normed space,βa bornology,f a continuous convex function on an open convex set DE. Then the following statements are equivalent:

(i) f is β-differentiable at xD.

(ii) Each selectionϕ:DE for the subdifferential∂f is τk·kβ contin- uous at xD.

(iii) There exists a selectionϕ:DE for the subdifferential∂fwhich is τk·kβ continuous at xD.

Proof. (i)⇒(ii). According to the previous proposition, ∂f is τk·kβ u.s.c., hence each of its selections will be τk·kβ continuous.

(ii)⇒(iii). This implication is obvious.

(iii)⇒(i). ForyD we have hϕ(x), y−xi ≤f(y)−f(x), because ϕ(x)

∂f(x). Usingϕ(y)∂f(x) one obtains hϕ(y), x−yi ≤f(x)−f(y),hence:

(9) 0≤f(y)f(x)− hϕ(x), y−xi ≤ hϕ(y)ϕ(x), yxi.

ForhE, t >0, replacing in (9)y=x+th, dividing bytand lettingt→0+, one obtains 0≤d+f(x)(h)−ϕ(x)(h)≤0,hence

d+f(x) =ϕ(x)E,

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and f is G-differentiable at x.

ForBβ,hB,t >0,y=x+th, we have

0≤ 1t(f(x+th)f(x))−df(x)≤ hϕ(x+th)ϕ(x), hi.

From the τk·kβ continuity of ϕ, the right hand side tends to 0 uniformly forhB (=bounded) as t→0+,and the conclusion follows.

REFERENCES

[1] Deville, R., Godefroy, G.and Zizler,V., Renormings and Smoothness in Banach Spaces,Monographs and Surveys in Pure and Appl. Math., Longman,64, 1993.

[2] Giles, J. R.,Convex Analysis with Application to Differentiation of Convex Functions, Research Notes in Math.,58, Pitman, 1982.

[3] Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, 2nd ed., Springer, 1993.

[4] Rainwater, J.,Yet more on the differentiability of convex functions,Proc. Amer. Math.

Soc.,103, no. 3, pp. 773–778, 1988.

[5] Siret¸chi, Gh.,Functional Analysis,Universitatea Bucure¸sti, 1982 (in Romanian).

[6] Zajicek, L.,On the differentiation of convex functions in finite and infinite dimensional spaces,Czechoslovak Math. J.,29, pp. 340–348, 1979.

Received by the editors: April 14, 2004.

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