View of On \(\beta\)-differentiability of norms

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 x ∈ U. A functionf :U →E₁ is said to be Gˆateaux differentiable atx if there exists a linear continuous mapping denoted df(x) :E →E1 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) : E → E₁ such that for each ε > 0, there existsδ >0 satisfying

(2) kf(x+h)−f(x)−f⁰(x)(h)k ≤εkhk, for each h∈B_E(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 U ⊆ E, 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) [=: d⁻f(x)(h)], for each h∈E.

∗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].

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 space`¹ 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 h ∈ B[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, B2 ∈ β there exists B3 ∈ β such that B₁, B₂ ⊆ B₃) 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 h ∈ B for each B ∈ β.

This turns out to be equivalent with the convergence in the uniform struc- ture F_β(E,R). We shall denote by τ_β 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: G ⊆ β ⊆ F, G ⊆ H ⊆ W ⊆ F; if f is β₂-differentiable and β₁ ⊆ β₂, 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 x∈U if and only if, for each B∈β, the limit

(3) lim

t→0+

1

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

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) = lim_t→0+¹_t(f(x−th)−f(x)), which hold uniformly for h ∈ B;

by addition the desired conclusion follows.

Sufficiency. Choose B ∈ β, 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(x−th)−2f(x) < tε, for each h ∈ B and t ∈(0, δ). (B is bounded, so, for sufficiently small δ >0 one has x±th∈B.)

We have

hx^∗, thi ≤f(x+th)−f(x), hx^∗,−thi ≤f(x−th)−f(x), and for 0< t < δ,h∈B one obtains:

0≤f(x+th)−f(x)− hx^∗, thi

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

≤εt+ 0 =εt,

which implies that (1) holds uniformly for h∈B.

Remark1. a)For the Fr´echet differentiability it is sufficient that the limit (3) holds uniformly on the unit sphere S_E; 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. Letf_n(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 series^P_n∈

N f_n 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(f_n(x+th) +f_n(x−th)−2f_n(x))

= ¹_t(f(x+th) +f(x−th)−2f(x))

(valid forx∈U,t >0,h∈B∈β provided that x+th∈U).

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 S_E of E.

The norm k·k isβ-differentiable atx if and only if the following condition holds: for all sequences x^∗_n, y^∗_n ∈ S_E^∗ satisfying x^∗_n(x) → 1, y_n^∗(x) → 1 one has x^∗_n−y_n^∗ →0 in F_β(E,R).

Proof. Necessity. Let B ∈ β, ε > 0, x^∗_n, y_n^∗ ∈ S_E^∗ satisfy x^∗_n(x) → 1, y_n^∗(x) → 1. Choosing B⁰ ∈ β 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 h∈B⁰.

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

|1−x^∗_n(x)|+|1−y_n^∗(x)|< εδ.

We have

x^∗_n(x+th) +y^∗_n(x−th)≤ kx+thk+kx−thk ≤2 +εδ, hence

x^∗_n(th)−y^∗_n(th)≤1−x^∗_n(x) + 1−y_n^∗(x) +εδ <2εδ, forn≥n0. By takingt=δ, one obtainsx^∗_n(h)−y_n^∗(h)<2ε, forn≥n₀,h∈B⁰.

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

|x^∗_n(h)−y_n^∗(h)|<2ε, for each n≥n₀, h∈B.

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

Choosing x^∗_n, y_n^∗ ∈ S_E^∗ such that x^∗_n(x+t_nh_n) ≥ kx+t_nh_nk − kt_nh_nk/n and y_n^∗(x−tnhn)≥ kx−tnhnk − kt_nhnk/n one obtains

1≥x^∗_n(x)

=x^∗_n(x+t_nh_n)−x^∗_n(t_nh_n)

≥ kx+t_nh_nk −^ktn_n^hnk − kt_nh_nk

≥1−^ktn_n^hnk −2kt_nh_nk, hence x^∗_n(x)→1.

Similarly, y^∗_n(x)→1.

Because

x^∗_n(x+tnhn) +y_n^∗(x−tnhn)≥2 +εtn−2^ktn_n^hnk, we have

x^∗_n(h_n)−y_n^∗(h_n)≥ε−2^kh_n^nk ≥ ₂^ε, forn≥n₀,

in contradiction withx^∗_n−y^∗_n→0 inF_β(E,R)

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 →2^Y is said to be τ1-τ2 upper semicontinuous (u.s.c.) atx ∈X, if for each subset W ∈ τ₂ containing T(x),there exists V ∈ τ₁ containing x such that T(V) =

∪{T(v) :v∈V} ⊆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 2^E∗, 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 τ_k·k (generated by the norm) on E, and the topology τ_β of the β-convergence on E^∗, where β is a bornology on E. We remind that τ_F = τ_k·k^∗, where F is the Fr´echet bornology (of all bounded subsets), andτ_Gis 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 → 2^E∗ a set-valued operator. Then, the following statements are equivalent:

(i) T is τ_k·k-τ_β upper semicontinuous at x.

(ii) For each W ∈τ_β, T(x)⊆W, (x_n)_n∈N⊆E with kx_n−xk →0, there existsn0 ∈N, such that for n≥n0, T(xn)⊆W.

(iii) For each W ∈ τ_β, 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 {x^∗₀}, the above conditions are also equivalent with

(iv) For each (xn)_n∈N⊆E, withkx_n−xk →0 it follows that

(4) lim

n→∞sup| hx^∗−x^∗₀, hi |:x^∗ ∈T(x_n), 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 D⊆E. Then the subdifferential ∂f :D→2^E∗ is a τ_k·k-τ_G upper semicontinuous operator.

Note that in the general case, ∂f will not be τ_k·k-τ_β 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=^P_n∈

N|x(n)|, and f :E →R, f(x) =kxk. For h∈E, 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 h∈E,i.e., x(n)6= 0,(n∈N).

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

Defining xp = (α1, α2, ..., αp,0,0...), i.e. xp(n) =αn forn≤p and xp(n) = 0 for n > p, we obviously have kx_p−xk →0.

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

x^∗_p∈E^∗, x^∗_p ≤d⁺f(xp), hence x^∗_p∈∂f(x_p).

On the other hand,kdf(x)−x^∗_pk^∗ = 1, so ∂f is not τk·k-τF 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 D ⊆ E, which is β-differentiable at x ∈ D.

Then the subdifferential ∂f : D → 2^E∗ is an τ_k·k-τ_β upper semicontinuous operator.

Proof. Suppose by contradiction that∂f is notτ_k·k-τ_β u.s.c. atx. Applying (iv) from Proposition 4 one obtains that there existx_n∈E, withkx_n−xk →0, ε >0,B∈β,hn∈B,x^∗_n∈∂f(xn), such that

| hx^∗_n−x^∗₀, h_ni |>2ε, (n∈N), wherex^∗₀= df(x).

ChoseB⁰∈β such that B∪(−B)⊆B⁰.

Interchanging if necessary hn with−h_n(∈B⁰), we will have (6) hx^∗_n−x^∗₀, h_ni>2ε.

From theβ-differentiability off atx, there existsδ >0, such thatB[x, δm]⊆ D, wherem >0 is chosen such thatB⁰⊆B[0, m], and

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

Hence

(7) f(x+th_n)−f(x)− hx^∗, th_ni ≤tε, (n∈N, t∈(0, δ]).

Using the fact that x^∗_n ∈ ∂f(xn), one obtains hx^∗_n, x+δhn−xni ≤ f(x+ δh_n)−f(x_n),hence

(8) hx^∗_n, δhni ≤f(x+δhn)−f(x) +hx^∗_n, xn−xni+f(x)−f(xn).

From (6), (7) and (8) we have 2εδ <hx^∗_n−x^∗₀, δhni

=hx^∗_n, δh_ni − hx^∗₀, δh_ni

≤f(x+δh_n)−f(x) +hx^∗_n, x_n−x_ni+f(x)−f(x_n)− hx^∗₀, δh_ni

=(f(x+δhn)−f(x)− hx^∗₀, δhni) +hx^∗_n, xn−xni+f(x)−f(xn)

≤εδ+kx^∗_nkkx_n−xnk+|f(x)−f(xn)|.

The convex function f being continuous, it is locally Lipschitz, hence the sequencekx^∗_nkis 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 D ⊆ E. Then the following statements are equivalent:

(i) f is β-differentiable at x∈D.

(ii) Each selectionϕ:D→E^∗ for the subdifferential∂f is τ_k·k-τ_β contin- uous at x∈D.

(iii) There exists a selectionϕ:D→E^∗ for the subdifferential∂fwhich is τ_k·k-τ_β continuous at x∈D.

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

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

(iii)⇒(i). Fory ∈D 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), y−xi.

Forh∈E, 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^∗,

and f is G-differentiable at x.

ForB ∈β,h∈B,t >0,y=x+th, we have

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

From the τ_k·k-τ_β continuity of ϕ, the right hand side tends to 0 uniformly forh∈B (=bounded) as t→0+,and the conclusion follows.

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Received by the editors: April 14, 2004.