**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 *E*1 be Banach spaces, *U* an open subset in *E* and *x* ∈ *U.* A
function*f* :*U* →*E*_{1} is said to be *Gˆateaux differentiable* at*x* if there exists a
linear continuous mapping denoted df(x) :*E* →*E*1 such that for each*h* in*E*
one has

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

*t→0+*

1

*t*(f(x+*th)*−*f*(x)).

The function*f* is said to be*Fr´echet differentiable*at*x*if there exists a linear
continuous mapping denoted *f*^{0}(x) : *E* → *E*_{1} such that for each *ε >* 0, there
exists*δ >*0 satisfying

(2) kf(x+*h)*−*f*(x)−*f*^{0}(x)(h)k ≤*εkhk,* for each *h*∈*B** _{E}*(x, δ).

The two linear mappings *df(x), f*^{0}(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. *E*1 =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 of*f* at*x*is 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 (x*n*)_{n∈}_{N} having only nonzero components.

It is well known that the function*f* is Fr´echet differentiable at*x*if 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.

Let*β* be a nonempty family of bounded sets in*E* whose union is *E,*which
is directed with respect to ⊆ (i.e., for each *B*1*, B*2 ∈ *β* there exists *B*3 ∈ *β*
such that *B*_{1}*, B*_{2} ⊆ *B*_{3}) and is invariant under scalar multiplication. Such a
family is named*bornology* in Phelps’ monograph [3].

The function *f* is said to be*β-differentiable* at the point *x* if*f* 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*

*the topology induced by this uniform structure.*

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

• *β*=*G*= the family of all finite subsets in*E* (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 in*E* (generating the
strong Hadamard differentiability).

One obviously has the inclusions: *G* ⊆ *β* ⊆ *F, G* ⊆ *H* ⊆ *W* ⊆ *F*; if *f*
is *β*_{2}-differentiable and *β*_{1} ⊆ *β*_{2}, then *f* is also *β*_{1}-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) = lim*t→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 pointx*0*, then each function* *f**n* *is* *β-differentiable at*
*x*_{0}*.*

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

0≤ ^{X}

*n∈*N
1

*t*(f* _{n}*(x+

*th) +f*

*(x−*

_{n}*th)*−2f

*(x))*

_{n}= ^{1}* _{t}*(f(x+

*th) +f(x*−

*th)*−2f(x))

(valid for*x*∈*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 the*β-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*

^{∗}

*(x) → 1,*

_{n}*y*

_{n}^{∗}(x) → 1

*one*

*has*

*x*

^{∗}

*−*

_{n}*y*

_{n}^{∗}→0

*in*

*F*

*(E,R).*

_{β}*Proof. Necessity.* Let *B* ∈ *β,* *ε >* 0, *x*^{∗}_{n}*, y*_{n}^{∗} ∈ *S*_{E}^{∗} satisfy *x*^{∗}* _{n}*(x) → 1,

*y*

_{n}^{∗}(x) → 1. Choosing

*B*

^{0}∈

*β*such that

*B*∪(−B) ⊆

*B*

^{0}and applying the preceding theorem, there exists

*δ >*0 such that for

*t*∈(0, δ] one has

kx+*thk*+kx−*thk<*2 +*εt*≤2 +*εδ,* for each *h*∈*B*^{0}*.*

The hypothesis implies the existence of a positive integer *n*0 such that for
*n*≥*n*_{0}:

|1−*x*^{∗}* _{n}*(x)|+|1−

*y*

_{n}^{∗}(x)|

*< εδ.*

We have

*x*^{∗}* _{n}*(x+

*th) +y*

^{∗}

*(x−*

_{n}*th)*≤ kx+

*thk*+kx−

*thk ≤*2 +

*εδ,*hence

*x*^{∗}* _{n}*(th)−

*y*

^{∗}

*(th)≤1−*

_{n}*x*

^{∗}

*(x) + 1−*

_{n}*y*

_{n}^{∗}(x) +

*εδ <*2εδ, for

*n*≥

*n*0

*.*By taking

*t*=

*δ, one obtainsx*

^{∗}

*(h)−*

_{n}*y*

_{n}^{∗}(h)

*<*2ε, for

*n*≥

*n*

_{0},

*h*∈

*B*

^{0}.

For*h*∈*B, we have* ±h∈*B*^{0}, and the last inequality implies

|x^{∗}* _{n}*(h)−

*y*

_{n}^{∗}(h)|

*<*2ε, for each

*n*≥

*n*

_{0}

*, 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*

*→ 0, kx+*

_{n}*t*

*n*

*h*

*n*k+kx−

*t*

*n*

*h*

*n*k ≥2 +

*εt*

*n*.

Choosing *x*^{∗}* _{n}*,

*y*

_{n}^{∗}∈

*S*

_{E}^{∗}such that

*x*

^{∗}

*(x+*

_{n}*t*

_{n}*h*

*) ≥ kx+*

_{n}*t*

_{n}*h*

*k − kt*

_{n}

_{n}*h*

*k/n and*

_{n}*y*

_{n}^{∗}(x−

*t*

*n*

*h*

*n*)≥ kx−

*t*

*n*

*h*

*n*k − kt

_{n}*h*

*n*k/n one obtains

1≥*x*^{∗}* _{n}*(x)

=*x*^{∗}* _{n}*(x+

*t*

_{n}*h*

*)−*

_{n}*x*

^{∗}

*(t*

_{n}

_{n}*h*

*)*

_{n}≥ kx+*t*_{n}*h** _{n}*k −

^{kt}

^{n}

_{n}

^{h}

^{n}^{k}− kt

_{n}*h*

*k*

_{n}≥1−^{kt}^{n}_{n}^{h}^{n}^{k} −2kt_{n}*h** _{n}*k,
hence

*x*

^{∗}

*(x)→1.*

_{n}Similarly, *y*^{∗}* _{n}*(x)→1.

Because

*x*^{∗}* _{n}*(x+

*t*

*n*

*h*

*n*) +

*y*

_{n}^{∗}(x−

*t*

*n*

*h*

*n*)≥2 +

*εt*

*n*−2

^{kt}

^{n}

_{n}

^{h}

^{n}^{k}

*,*we have

*x*^{∗}* _{n}*(h

*)−*

_{n}*y*

_{n}^{∗}(h

*)≥*

_{n}*ε*−2

^{kh}

_{n}

^{n}^{k}≥

_{2}

^{ε}*,*for

*n*≥

*n*

_{0}

*,*

in contradiction with*x*^{∗}* _{n}*−

*y*

^{∗}

*→0 in*

_{n}*F*

*(E,R)*

_{β}If*f* is a continuous convex function on an open convex subset*U* 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 of*f; in the same time such results motivated an intensive research*
on the set-valued operators.

If (X, τ_{1} ), (Y, τ_{2} ) are topological spaces, a set-valued operator*T* :*X* →2* ^{Y}*
is said to be

*τ*1-τ2

*upper semicontinuous (u.s.c.)*at

*x*∈

*X, if for each subset*

*W*∈

*τ*

_{2}containing

*T*(x),there exists

*V*∈

*τ*

_{1}containing

*x*such that

*T*(V) =

∪{T(v) :*v*∈*V*} ⊆*W*.

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

We are interested in the case when the operator*T* acts between*E* 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), and

*τ*

*is the weak*

_{G}^{∗}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

*existsn*0 ∈N

*, such that for*

*n*≥

*n*0

*,*

*T*(x

*n*)⊆

*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* (x*n*)_{n∈}_{N}⊆*E, with*kx* _{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+*diam*T*(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

lim*t→0*

|x(n)+th(n)|−|x(n)|

*t*

= ^{X}

*n∈*N,x(n)6=0

(sign*x(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* *x**p* = (α1*, α*2*, ..., α**p**,*0,0...), i.e. *x**p*(n) =*α**n* *forn*≤*p* *and* *x**p*(n) =
0 *for* *n > p,* *we obviously have* kx* _{p}*−

*xk →*0.

*But* d^{+}*f*(x*p*)(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*

^{∗}

*≤d*

_{p}^{+}

*f*(x

*p*),

*hence*

*x*

^{∗}

*∈*

_{p}*∂f(x*

*).*

_{p}*On the other hand,*kdf(x)−*x*^{∗}* _{p}*k

^{∗}= 1, so

*∂f*

*is not*

*τ*k·k

*-τ*

*F*

*u.s.c. at*

*x*

*(cf.*

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

In this example,*f* is not*F-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 not*τ*_{k·k}-τ* _{β}* u.s.c. at

*x. Applying*(iv) from Proposition 4 one obtains that there exist

*x*

*∈*

_{n}*E, with*kx

*−xk →0,*

_{n}*ε >*0,

*B*∈

*β,h*

*n*∈

*B,x*

^{∗}

*∈*

_{n}*∂f(x*

*n*), such that

| hx^{∗}* _{n}*−

*x*

^{∗}

_{0}

*, h*

*i |*

_{n}*>*2ε, (n∈N), where

*x*

^{∗}

_{0}= df(x).

Chose*B*^{0}∈*β* such that *B*∪(−B)⊆*B*^{0}.

Interchanging if necessary *h**n* with−h* _{n}*(∈

*B*

^{0}), we will have (6) hx

^{∗}

*−*

_{n}*x*

^{∗}

_{0}

*, h*

*i*

_{n}*>*2ε.

From the*β-differentiability off* at*x, there existsδ >*0, such that*B*[x, δm]⊆
*D, wherem >*0 is chosen such that*B*^{0}⊆*B[0, m], and*

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

Hence

(7) *f(x*+*th** _{n}*)−

*f*(x)− hx

^{∗}

*, th*

*i ≤*

_{n}*tε,*(n∈N

*, t*∈(0, δ]).

Using the fact that *x*^{∗}* _{n}* ∈

*∂f(x*

*n*), one obtains hx

^{∗}

_{n}*, x*+

*δh*

*n*−

*x*

*n*i ≤

*f*(x+

*δh*

*)−*

_{n}*f*(x

*),hence*

_{n}(8) hx^{∗}_{n}*, δh**n*i ≤*f(x*+*δh**n*)−*f*(x) +hx^{∗}_{n}*, x**n*−*x**n*i+*f(x)*−*f*(x*n*).

From (6), (7) and (8) we have
2εδ <hx^{∗}* _{n}*−

*x*

^{∗}

_{0}

*, δh*

*n*i

=hx^{∗}_{n}*, δh** _{n}*i − hx

^{∗}

_{0}

*, δh*

*i*

_{n}≤f(x+*δh** _{n}*)−

*f(x) +*hx

^{∗}

_{n}*, x*

*−*

_{n}*x*

*i+*

_{n}*f*(x)−

*f(x*

*)− hx*

_{n}^{∗}

_{0}

*, δh*

*i*

_{n}=(f(x+*δh**n*)−*f*(x)− hx^{∗}_{0}*, δh**n*i) +hx^{∗}_{n}*, x**n*−*x**n*i+*f*(x)−*f(x**n*)

≤εδ+kx^{∗}* _{n}*kkx

*−*

_{n}*x*

*n*k+|f(x)−

*f*(x

*n*)|.

The convex function *f* being continuous, it is locally Lipschitz, hence the
sequencekx^{∗}* _{n}*kis bounded (by the Lipschitz constant). For

*n*→ ∞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 *τ*k·k-τ*β* u.s.c.,
hence each of its selections will be *τ*_{k·k}-τ* _{β}* continuous.

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

(iii)⇒(i). For*y* ∈*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.*

For*h*∈*E, t >*0, replacing in (9)*y*=*x*+*th, dividing byt*and letting*t*→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.*

For*B* ∈*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 for

*h*∈

*B*(=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.