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∈BE(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 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: 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) = limt→0+1t(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 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(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 SE of E.
The norm k·k isβ-differentiable atx if and only if the following condition holds: for all sequences x∗n, y∗n ∈ SE∗ satisfying x∗n(x) → 1, yn∗(x) → 1 one has x∗n−yn∗ →0 in Fβ(E,R).
Proof. Necessity. Let B ∈ β, ε > 0, x∗n, yn∗ ∈ SE∗ satisfy x∗n(x) → 1, yn∗(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−yn∗(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−yn∗(x) +εδ <2εδ, forn≥n0. By takingt=δ, one obtainsx∗n(h)−yn∗(h)<2ε, forn≥n0,h∈B0.
Forh∈B, we have ±h∈B0, and the last inequality implies
|x∗n(h)−yn∗(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,hn ∈ B\{0},tn >0 such that tn → 0, kx+tnhnk+kx−tnhnk ≥2 +εtn.
Choosing x∗n, yn∗ ∈ SE∗ such that x∗n(x+tnhn) ≥ kx+tnhnk − ktnhnk/n and yn∗(x−tnhn)≥ kx−tnhnk − ktnhnk/n one obtains
1≥x∗n(x)
=x∗n(x+tnhn)−x∗n(tnhn)
≥ kx+tnhnk −ktnnhnk − ktnhnk
≥1−ktnnhnk −2ktnhnk, hence x∗n(x)→1.
Similarly, y∗n(x)→1.
Because
x∗n(x+tnhn) +yn∗(x−tnhn)≥2 +εtn−2ktnnhnk, we have
x∗n(hn)−yn∗(hn)≥ε−2khnnk ≥ 2ε, forn≥n0,
in contradiction withx∗n−y∗n→0 inFβ(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 →2Y 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 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·k-τβ upper semicontinuous at x.
(ii) For each W ∈τβ, T(x)⊆W, (xn)n∈N⊆E with kxn−xk →0, there existsn0 ∈N, such that for n≥n0, T(xn)⊆W.
(iii) For each W ∈ τβ, 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, withkxn−xk →0 it follows that
(4) lim
n→∞sup| hx∗−x∗0, hi |:x∗ ∈T(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 D⊆E. Then the subdifferential ∂f :D→2E∗ is a τk·k-τG upper semicontinuous operator.
Note that in the general case, ∂f will not be τk·k-τβ 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 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) =αn>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 kxp−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(xp).
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 → 2E∗ is an τk·k-τβ upper semicontinuous operator.
Proof. Suppose by contradiction that∂f is notτk·k-τβ u.s.c. atx. Applying (iv) from Proposition 4 one obtains that there existxn∈E, withkxn−xk →0, ε >0,B∈β,hn∈B,x∗n∈∂f(xn), such that
| hx∗n−x∗0, hni |>2ε, (n∈N), wherex∗0= df(x).
ChoseB0∈β such that B∪(−B)⊆B0.
Interchanging if necessary hn with−hn(∈B0), we will have (6) hx∗n−x∗0, hni>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+thn)−f(x)− hx∗, thni ≤tε, (n∈N, t∈(0, δ]).
Using the fact that x∗n ∈ ∂f(xn), one obtains hx∗n, x+δhn−xni ≤ f(x+ δhn)−f(xn),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, δhni − hx∗0, δhni
≤f(x+δhn)−f(x) +hx∗n, xn−xni+f(x)−f(xn)− hx∗0, δhni
=(f(x+δhn)−f(x)− hx∗0, δhni) +hx∗n, xn−xni+f(x)−f(xn)
≤εδ+kx∗nkkxn−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 τk·k-τβ contin- uous at x∈D.
(iii) There exists a selectionϕ:D→E∗ for the subdifferential∂fwhich is τk·k-τβ 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 τk·k-τβ 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≤ 1t(f(x+th)−f(x))−df(x)≤ hϕ(x+th)−ϕ(x), hi.
From the τk·k-τβ 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.