Rev. R. Acad. Cien. Serie A. Mat.
SAM
VOL.99(2), 2005, pp. 167–181
An´alisis Matem´atico / Mathematical Analysis
A One-sided Version of Alexiewicz-Orlicz’s Differentiability Theorem
E. Corbacho, A. Plichko and V. Tarieladze
Abstract. Modifying appropriately the method of a forgotten work [1], we show that if a continuous mapping from a nonempty open subsetU of a metrizable separable Baire topological vector spaceE to a locally convex space is directionally right differentiable onU along a comeager subset ofE, then it is generically Gˆateaux differentiable onU. Our result implies that every metrizable separable Baire topological vector space is weak Asplund. It partially covers also the similar statements of [8, 16, 23].
Una version lateral del teorema de diferenciabilidad de Alexiewicz-Orlicz Resumen. Modificando adecuadamente el m´etodo de un trabajo olvidado [1], probamos que si una aplicaci´on continua, de un subconjunto abierto no vac´ıoUde un espacio vectorial topol´ogico metrizable separable y de BaireE, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha enU seg´un un subconjunto comagro deE, entonces, es gen´ericamente Gˆateaux diferenciable enU. Nuestro resultado implica que cualquier espacio vectorial topol´ogico, metrizable, separable y de Baire, es d´ebilmente Asplund. Tambi´en cubre parcialmente similares resultados de [8, 16, 23].
1 Introduction
The main goal of the present article is to show that in a rather general setting the one-sided directional differentiability of a mapping already implies its generic Gˆateaux differentiability. A precise formulation of the obtained result and clarification of its connection with other known related statements will be easier to give after recalling the corresponding definitions.
We consider vector spaces over the same fieldKof realRor complexCnumbers and keep the following notations:
— Eis a topological vector space,U,Ha non-empty subsets ofEbeingU open,
— Fa Hausdorff topological vector space,
— f:U →Fa function.
Moreover, for a fixed scalart∈K\ {0}andx∈U we introduce a mapping∆f,x,t: E→F, defined at a givenh∈Eas follows:
∆f,x,t(h) =
f(x+th)−f(x)
t , whenx+th∈U θ, whenx+th6∈U.
Presentado por M. Valdivia.
Recibido: 20 de noviembre de 2005.Aceptado: 7 de diciembre de 2005.
Palabras clave / Keywords: Gˆateaux derivative, one-sided directional derivative Mathematics Subject Classifications: 46G05; 26E15
c
2005 Real Academia de Ciencias, Espa˜na.
We adopt the following definitions (cf. [4, 20, 28]):
Definition 1 The functionf will be caled:
— Directionally differentiable atxalong a vectorh∈Eif the functiont7→∆f,x,t(h)has a limit inF ast→0.
— Directionally differentiable atxalongH iffis directionally differentiable atxalong everyh∈H.
— Directionally differentiable atxiff is directionally differentiable atxalong the whole spaceE.
— Directionally right differentiable atxalong a vectorh∈Eif the functiont7→∆f,x,t(h)has a limit inF ast∈R, t >0andt→0.
— Directionally right differentiable at xalongH if f is directionally right differentiable atxalong everyh∈H.
— Directionally right differentiable or one-sided directionally differentiable atxif f is directionally right differentiable atxalong the whole spaceE.
Iffis directionally differentiable atxalongH, then the mappingDf,x:H →F defined forh∈Hby the equality
Df,x(h) = lim
t→0∆f,x,t(h) is called thedirectional derivative off atxalongH.
Clearly, iffis directionally differentiable atxthen its directional derivative atxis defined everywhere onE, i.e.Df,x:E→F.
Iff is directionally right differentiable at xalong H, then the mapping Df,x+ : H → F defined for h∈Hby the equality
D+f,x(h) = lim
t∈R, t>0, t→0∆f,x,t(h)
is called thedirectional right derivative or one-sided directional derivative off atxalongH.
Iff is directionally right differentiable atx, then its directional right derivative atxis defined every- where onE, i.e.Df,x+ : E→F.
Definition 2 The functionf is said to be Gˆateaux differentiable atxiffis directionally differentiable at xand its directional derivativeDf,x:E→Fis a continuous linear mapping.
Definition 3 LetE,F be normed spaces. The functionf is said to be Fr´echet differentiable atxiff is directionally differentiable atx,
sup
h∈E,khk=1
k∆f,x,t(h)−Df,x(h)k →0 as t→0 and the directional derivativeDf,xis a continuous linear mapping.
Iff is Fr´echet differentiable at x, thenf is continuous and Gˆateaux differentiable atx. A Gˆateaux differentiable atxfunction may not be continuous at x; also, a continuous Gˆateaux differentiable at x function may not be Fr´echet differentiable atx( even ifU =R2andF =R).
Let us recall that a set in a topological space which is a countable union of nowhere dense sets is called meager(orof the first category) and the complement of a meager set is calledcomeager(orresidual). A topological spaceXis called aBaire spaceif the intersection of any sequence of open dense subsets ofXis dense. IfX is a non-empty Baire space, then the denseGδsubsets ofX are not meager and any comeager subset ofX contains a denseGδ set. A comeager subset of a Baire space is a Baire space as well (see, e.g., [10, Ch.IX,§5, Proposition 5]). The Baire-Hausdorff theorem asserts that ifX is either a completely
metrizable or a locally compact regular topological space, then X is Baire. There are also metrizable separable Baire topological spaces which are neither completely metrizable, nor locally compact.
The mappingf:U →Fis calledgenerically Gˆateaux differentiableonUif it is Gˆateaux differentiable at every point of a denseGδsubset ofU.
A topological vector spaceF will be calleddually separatedif for everyy ∈ F \ {0} there exists a continuous linear functionalu:F →Ksuch thatu(y)6= 0. Every Hausdorff locally convex space is dually separated. If0 < p < 1, then the sequence spacelpwith its standard topology presents an example of a non-locally convex complete separable metrizable dually separated space.
Our starting point was the following remarkable statement obtained in [1].
(AO) IfE is areal separable Banach space, F is a Banach space, f:U → F is continuous andf is directionally differentiable at everyx∈U, thenf is generically Gˆateaux differentiable onU(see [1, Theorem]).
In a huge amount of the literature dedicated to the differentiability questions of mappings between Banach and more general spaces, seemingly only [30] contains [1] in its referencee’s list. However, even in [30] the statement (AO) is not mentioned at all.
In the given article we show that, modifying appropriately the method of [1], it is possible to obtain the next one-sided version of (AO).
Theorem 1 LetEbe a real separable metrizable Baire topological vector space,U ⊂Ebe an open set, Fbe a metrizable topological vector space andf:U →F be a continuous mapping.
(a) IfF is locally convex andf is directionally right differentiable at everyx∈ U along a comeager subsetHof an open neighborhood of zeroV ⊂E, thenfis generically Gˆateaux differentiable onU. (b) IfFis a (not necessarily locally convex) dually separated space andfis directionally right differen-
tiable at everyx∈U along wholeE, thenfis generically Gˆateaux differentiable onU.
It is clear that Theorem 1 contains and refines (AO). In the next remark we discuss a consequence and some results related with Theorem 1 in case whenE=R.
Remark 1 LetU be an open subset ofR. We note:
— f is directionally differentiable atxalong1ifff is differentiable atxin ordinary sense. Moreover, iff is directionally differentiable atxalong1, thenDf,x(1) = f0(x), wheref0(x)stands for the ordinary derivative off atx.
— f is directionally right differentiable atxalong1ifff is right differentiable atxin ordinary sense.
Moreover, iff is directionally right differentiable atxalong1, thenD+f,x(1) =f+0 (x), wheref+0(x) stands for the ordinary right derivative off atx.
— f is directionally right differentiable atxalong−1ifff is left differentiable atxin ordinary sense.
Moreover, iff is directionally right differentiable atxalong−1, thenD+f,x(−1) =−f−0(x), where f−0(x)stands for the ordinary left derivative offatx.
Taking into account these observations, from Theorem 1 we get the next statement:
(I)IfF is a metrizable locally convex space andf:U →F is a continuous function which is right and left differentiable at each pointx∈ U andf+0,f−0 are continuous functions onU, then there exists some comeager subsetC⊂U, such thatf is differentiable at every pointx∈C.
Note, however, that the following stronger version of (I) (which is not a direct consequence of Theo- rem 1) is true:
(II)IfF is a Banach space andf:U →F is left and right differentiable at each pointx ∈U, then there exists some finite or countable subsetS ⊂U, such thatf is differentiable at every pointx∈U\S
(see [14, Ch.VIII,§.4, Probleme 2 (b)]; in [2, p. 159, Lemma 1] a similar statement is proved for real-valued functions and it is attributed to W. Sierpinski).
In the next remark are collected some other related results and consequences of Theorem 1.
Remark 2
a) G ˆateaux differentiability of convex functions
(J1) IfUis an open convex subset ofRandf:U →Ris a convex function, thenf is continuous onU, is right and left differentiable at each point ofU and the set of points of non-differentiability off is at most countable (see, e.g.,[26, p. 9, Th. 1.16]).
(J2) IfUis a convex open subset of a real topological vector space andf:U →Ris a convex functional, thenf is directionally right differentiable at each point ofU(see, e.g.,[26, p. 2, Lemma 1.2]).
(M) IfU is a convex open subset of a real separable Banach spaceE andf: U → Ris a continuous convex functional, thenf is generically Gˆateaux differentiable onU (this is theMazur’s theorem, see[24]; see also[26, p. 12, Th. 1.20]).
A topological vector spaceE is said to be aweak Asplund space provided every continuous convex function defined on a nonempty open convex subset U ofE is generically Gˆateaux differentiable on U (cf.[26, p. 13, Def. 1.22]and[17, Def. 1.01]).
Theorem1together with(J2), implies the next generalization of(M):
(M1) IfEis a real separable metrizable Baire topological vector space, thenEis a weak Asplund space.
In particular, we get that every separable Baire normed space is weak Asplund. Whether every separa- ble weak Asplund normed space is Baire, we do not know.
Note, however, that for some non-locally convex complete separable metrizable spacesEthe conclusion of (M1)may hold trivially (e.g., it is known that ifE = Lp[0,1]with0 < p < 1andf: E → Ris a continuous convex functional, thenfis a constant functional).
(M2) An analogue of(M)may fail in general: the norm ofE=L∞[0,1]is nowhere Gˆateaux differentiable (this was noticed already in[24]).
From(M2), since the Banach spacesE =L∞[0,1]andl∞are isomorphic, we get thatl∞admits an equivalent norm which is nowhere Gˆateaux differentiable (see also[26, Example 1.21]for a simple example of a nowhere Gˆateaux differentiable continuous seminorm given onl∞).
It follows from(J2)and(M2)that Theorem1(b)may fail whenEis a non-separable Banach space and F = R(we do not know whether or not some version of (AO)remains true whenEis a non-separable Banach space).
In connection with(M2)let us mention that the class of weak Asplund spaces is much wider than the class of separable Banach spaces as the next generalization of Mazur’s theorem shows:
(As1) IfE is a (not necessarily separable) closed subspace of a WCG Banach space, thenE is a weak Asplund space ([3, Th. 2]; see also[26, p. 37, Th. 2.45]and[17, Th. 1.3.4]).
Whether every WCG Baire normed space is weak Asplund, we do not know.
b) Fr ´echet differentiability of convex functions
It is known that if a continuous convexf: U → Rgiven an open convex subsetU a finite-dimensional Banach spaceEis Gˆateaux differentiable at a pointx∈U, then it is Fr´echet differentiable atx. A similar assertion is not true in general as the next result shows.
(BF) IfE is an infinite-dimensional real Banach space, then there exists an equivalent normf : E →R and a pointx∈E, such thatf is Gˆateaux differentiable atx, butfis not Fr´echet differentiable atx (see[9, Theorem 1]).
A normed spaceE is said to be anAsplund spaceprovided every continuous convex function defined on a nonempty open convex subsetU ofEis Fr´echet differentiable at each point of someGδ subset ofU (see[26, p. 13, Def. 1.22]; cf. also[17, Def. 1.01]).
(As2) A Banach spaceEis an Asplund space iff every separable closed subspace ofEhas a separable dual (cf.[3, Th. 1]; see also[26, p. 32, Th. 2.34]and[17, Th. 1.1.1]).
We refer to [7, 13, 17, 18, 26]for an extensive study of the differentiability problems of convex and Lipschitz functions given on a Banach space.
c) Differentiability in complex spaces
(Sukh) IfEis acomplexBanach space andf:U →Cis locally bounded and is directionally differentiable at everyx∈U, thenfis Fr´echet differentiable at everyx∈U(see[28, Th. 10]).
(Z1) IfE,F arecomplexBanach spaces andf:U →F is directionally differentiable at everyx∈ U, then for everyx∈U the directional derivativeDf,x:E→Fis linear (see[32, (2.3)]).
WhenEis infinite-dimensional, in(Z1), in general, it cannot be asserted the continuity ofDf,x: E→F (because, iff:E→Cis a non-continuous linear functional, thenDf,x=f,∀x∈E).
(Z2) IfE, F arecomplex Banach spaces, the restriction off:U → F on some comeager subset ofU is continuous and the functionf is directionally differentiable at everyx ∈ U, then f is Fr´echet differentiable at everyx∈U (see[32, (4.10)]).
An analogue of (Sukh)(and hence, of (Z2)too) is not true for real spaces: a continuous function f:R2→Rmay be Gˆateaux differentiable at everyx∈R2, but may not be Fr´echet differentiable at every x∈R2.
We can consider(AO)and Theorem1as a real continuous separable and category version of (Z2).
d) Other results related with Theorem 1
The next statement can be considered as a refinement of(AO)
(LW) IfEis real separable Banach space,Fis a Banach space,f is continuous onU and there is a dense GδsubsetBofU such that at everyx∈Bthe functionf is directionally differentiable along aGδ subsetH ⊂Ewhich is dense in a non-empty open subset ofE, then there exists a denseGδ subset AofBsuch thatf is Gˆateaux differentiable at everyx∈A(see[23, Th. 3.1]; cf also,[8, Coroll. 2 to Th. 1]).
In[16]it was proved the next one-sided version of (LW):
(Fa) LetEbe a separable Banach space,F a Banach space,U,V be nonempty open subsets ofEand f:U →F be a continuous mapping. Suppose that there exists a denseGδ subsetSofU×V such that∀(x, h)∈Sthe functiont7→∆f,x,t(h)has a limit inF ast >0andt→0.
Then there exists a denseGδ subsetAofU such thatf is Gˆateaux differentiable at everyx ∈ A (see[16, Th. 3.1]).
In[31]is contained the next version of (Fa):
(Zh) IfEis a (not necessarily separable) WCG Banach space,f:U → Ris continuous and there is a denseGδ subsetB ofU such thatf is directionally right differentiable at everyx∈ B, then there exists a denseGδsetA⊂Bsuch thatfis Gˆateaux differentiable at everyx∈A(see[31, Th. 4.4]).
Our Theorem1partially covers(LW)and(Fa).
The one sided directional differentiability was considered in [2, 19, 27] and many other works.
2 Auxiliary results and proofs
The following easily provable observations will be used later.
(S1) Iff is directionally differentiable atxalongh ∈ E andt ∈ Kis a fixed scalar,f is directionally differentiable atxalong the vectorthandDf,x(th) =tDf,x(h).
(S2) Iffis directionally right differentiable atxalongh∈Eandt∈R, t≥0is a fixed scalar, thenfis directionally right differentiable atxalong the vectorthandDf,x+ (th) =tD+f,x(h).
(S3) Iff is directionally differentiable atxalongh∈ E, thenf is directionally right differentiable atx along the vectorhandDf,x(h) =D+f,x(h).
(S4) Ifh ∈ E is a vector such thatf is directionally right differentiable at xalong the set {ζh : ζ ∈ K, |ζ|= 1}andD+f,x(ζh) =ζDf,x+ (h) ∀ζ ∈K, |ζ|= 1, thenf is directionally differentiable at xalonghandDf,x(h) =Df,x+ (h).
(S5) IfE,F are vector spaces overR, the setH is an additive subgroup ofE,f is directionally right differentiable atxalongH andD+f,x(h1+h2) = Df,x+ (h1) +Df,x+ (h1) ∀h1, h2 ∈ H, thenf is directionally differentiable atxalongHandDf,x(h) =D+f,x(h) ∀h∈H.
(S6) Iff is directionally (right) differentiable atxalongh∈Eanduis a continuous linear mapping from Fto a topological vector spaceG, thenu◦f:U →Gis directionally (right) differentiable atxalong handDu◦f,x(h) =u(Df,x(h))(Du◦f,x+ (h) =u(D+f,x(h))).
To present a proof Theorem 1, we need several statements which may have an independent interest. The next Lemma is a one-sided version of Lagrange’s mean value theorem.
Lemma 1 Lett1, t2∈R,t1< t2andϕ: [t1, t2]→Rbe a function satisfying the conditions:
(c1) ϕis bounded and attains its supremum and infimum on[t1, t2].
(c2) ϕis right differentiable at every point of]t1, t2[.
(c3) ϕis continuous att1. Then:
(a) There existλ∈[0,1]ands1, s2∈]t1, t2[such that ϕ(t2)−ϕ(t1)
t2−t1
=λϕ0+(s1) + (1−λ)ϕ0+(s2)
(b) There existss∈]t1, t2[such that
|ϕ(t2)−ϕ(t1)|
t2−t1 ≤ |ϕ0+(s)|.
(c) If ϕ0+(t) ≥ 0, ∀t ∈]t1, t2[(resp. if ϕ0+(t) ≤ 0, ∀t ∈]t1, t2[), then ϕis increasing (resp. is decreasing) on[t1, t2].
PROOF. We will prove first the next
Step 1.Ifϕsatisfies (c1) and is right differentiable at every point of[t1, t2[, then (a’) There existλ∈[0,1]ands1, s2∈[t1, t2[such that ϕ(t2)−ϕ(t1)
t2−t1
=λϕ0+(s1) + (1−λ)ϕ0+(s2).
(c’) If ϕ0+(t) ≥ 0, ∀t ∈ [t1, t2[(resp. if ϕ0+(t) ≤ 0, ∀t ∈]t1, t2[), then ϕis increasing (resp. is decreasing) on[t1, t2].
Proof of Step 1. (a’). Lett1 = 0,t2 = 1andϕ(1)−ϕ(0) = 0. In this case(a’)will be proved if we can see that
∃s1, s2∈[0,1[ ϕ0+(s1)≤0≤ϕ0+(s2). (1) Sinceϕ(1) =ϕ(0), it follows easily from(c1)that
∃α, β∈[0,1[ ϕ(α)≤ϕ(t)≤ϕ(β), ∀t∈[0,1]. (2) If we take now anyα, β∈[0,1[for which (2) holds, then (1) will be true withs2=βands1=α.
The conclusion of(a’)when[t1, t2]6= [0,1]follows from the already proved step, applied to the function ψ: [0,1]→Rgiven for at∈[0,1]by equality:
ψ(t) :=ϕ(t1+t(t2−t1)) t2−t1
−ϕ(t2)−ϕ(t1) t2−t1
t.
Since(c’)follows from(a’), Step 1 is proved.
Step 2.Proof of (a).Lett1= 0,t2= 1andϕ(1)−ϕ(0) = 0. In this case(a’)will be proved if we can see that
∃s1, s2∈[0,1[ ϕ0+(s1)≤0≤ϕ0+(s2). (3) Take anyα, β∈[0,1[for which (2) holds.
Ifα, β∈]0,1[, then (3) will be true withs2=βands1=α.
It remains to prove (3) in remaining cases:
Case 1:α=β = 0,
Case 2:α= 0and0< β <1, Case 3:β= 0and0< α <1.
In case 1 the functionϕis identically zero, and so, (3) holds trivially. Therefore we can exclude this case and can suppose thatϕ(α)< ϕ(β).
Consider Case 2. Fors1 := β, we haveϕ0+(s1) ≤0. So, we need to find somes2 ∈]0,1[such that ϕ0+(s2)≥0. Suppose this is not possible; then we shall haveϕ0+(s)≤0∀s∈]0,1[. Fix a numberr∈]0, β[.
Since(c2)is satisfied,ϕis right differentiable at each point of[r, β[. An application of Step 1(c’)for the restriction ofϕto[r, β], together withϕ0+(s)≤0 ∀s∈[r, β[, gives thatϕis decreasing on[r, β]. Hence, ϕ(r)≥ϕ(β), ∀r∈]0, β[. From this,since (c3) is satisfied att1= 0, we get
ϕ(0) = lim
r>0, r→0ϕ(r)≥ϕ(β),
but this contradicts to the inequalityϕ(0) =ϕ(α)< ϕ(β). Consequently, the relationϕ0+(s)≤0 ∀s∈ ]0,1[cannot hold, hence somes2∈]0,1[we haveϕ0+(s2)≥0and (3) is proved in Case 2.
The validity of (3) in Case 3 can be verified in a similar way.
The conclusion of(a)when[t1, t2]6= [0,1]follows from the already proved step, applied to the function ψ: [0,1]→Rgiven for at∈[0,1]by equality:
ψ(t) :=ϕ(t1+t(t2−t1)) t2−t1
−ϕ(t2)−ϕ(t1) t2−t1
t.
(b)and(c)follow from(a).
Remark 3 (1) Lemma1(a) in case of a continuous ϕis contained in [15, 16.3]; as L. Maligranga informed us, it dues, probably, to D. Kurepa.
(2) The ordinary mean value theorem is not a consequence of Lemma1(a), because, even ifϕ: [0,1]→R is continuous and it is right differentiable at every point of[0,1[, there may not exist a points∈[0,1[
with property:ϕ(1)−ϕ(0) =ϕ0+(s).
(3) Lemma1(c) in case of a continuousϕis known, see[12, Cor. 3.2.1]. In[11, Ch. I,§2, Prop. 2]is contained the next more general result:
(PrBo) If for acontinuousϕ: [t1, t2] → Rthere exists an at most countable setA ⊂ [t1, t2]such thatϕ is right differentiable at each point of[t1, t2]\Aandϕ0+(s) ≥ 0 for everys ∈ [t1, t2]\A, then ϕ(t1)≤ϕ(t2).
In (PrBo) “an at most countable setA ⊂ [t1, t2]” cannot be replaced by “a nowhere dense setA ⊂ [t1, t2]” (see[11, Ch. I,§2, p. 22, Rem. 2]).
Lemma 2 Let(F,k · k)be a normed space andϕ: [0,1] → F be a continuous function which is right differentiable at every point of[0,1[.
(a) (cf.[12, Cor. 3.2.2]) There existss∈]0,1[such thatkϕ(1)−ϕ(0)k ≤ kϕ0+(s)k.
(b) For everyr∈[0,1[there existss∈]0,1[such that
kϕ(1)−ϕ(0)−ϕ0+(r)| ≤ kϕ0+(s)−ϕ0+(r)k
PROOF. (a). By Hahn-Banach theorem for some continuous linear functionalu:F →Rwithkuk ≤ 1we havekϕ(1)−ϕ(0)k=u(ϕ(1)−ϕ(0)). An application of Lemma 1(b)to the functionu◦ϕgives(a).
(b). Fixr∈[0,1[and apply(a)to the new functiont7→ψ(t) :=ϕ(t)−ϕ0+(r)t.
Remark 4 The following stronger version of Lemma2(a) is known: if a continuousϕ: [0,1]→F there exists some finite or countable subsetA ⊂ [0,1], such that ϕis right differentiable at every point t ∈ [0,1[\A, then there existss∈]0,1]\Asuch thatkϕ(1)−ϕ(0)k ≤ kϕ0+(s)k(see[11, Ch. I,§2, exer. 14], or[14, Ch. VIII,§5, Prob. 7]).
Lemma 3 LetEbe a real Hausdorff topological vector space,U ⊂ E be an open set,F be a normed space,f:U →Fbe a mapping and the elementsb∈U,h∈Ebe such that
{b+th:t∈[0,1]} ⊂U. (4)
If either
(crd) the mappingf iscontinuousand it is directionally right differentiable at every x ∈ U along the vectorh,
or
(dd) the mappingf isdirectionally differentiableat everyx ∈ U along the vectorh, then there exists s∈]0,1[such that
kf(b+h)−f(b)−Df,b+ (h)k ≤ kD+f,b+sh(h)−Df,b+ (h)k. (5) PROOF. WriteA = {t ∈ R : b+th ∈ U}. SinceU is open in E, the setAis an open inRand since (4) is satisfied,[0,1]⊂A. Considerϕ:A→Fgiven fort∈Aby the equality:ϕ(t) =f(b+th).
If(crd)is satisfied, thenϕ:A→Fis continuous, right differentiable at everyt∈Aand ϕ0+(t) =D+f,b+th(h), ∀t∈A.
Now an application of Lemma 2(b)for the restriction ofϕto[0,1]⊂Agives (5).
If(dd)is satisfied, thenϕ:A→Fdifferentiableat everyt∈Aand ϕ0(t) =Df,b+th(h), ∀t∈A.
Since the differentiability ofϕimplies its continuity onA, again an application of Lemma 2(b)for the restriction ofϕto[0,1]⊂Agives (5).
The next statement is a one-sided version of [1, Lemma].
Proposition 1 LetEbe a real Hausdorff topological vector space,n >1a natural number,h1, . . . , hn∈ E,U ⊂Ebe an open set,x0∈U,F be a Hausdorff topological topological vector space andf:U →F be a continuous mapping.
(a) IfFis locally convex,fis directionally right differentiable at everyx∈Ualong the set{h1, . . . , hn} and the mapsx→ D+f,x(hi),i = 1, . . . , n−1are continuous atx0, thenf is directionally right differentiable atx0along the vectorh1+· · ·+hnand
D+f,x
0(h1+· · ·+hn) =D+f,x
0(h1) +· · ·+Df,x+
0(hn). (6)
(b) IfFis a (not necessarily locally convex) dually separated space,fis directionally right differentiable at everyx∈U along wholeEand the mapsx→Df,x+ (hi),i= 1, . . . , n−1are continuous atx0, then equality(6)holds.
(a’,b’) The conclusion of (a) (resp. of (b)) remains true for a not necessarily continuousf:U →Fprovided f isdirectionally differentiable at every x ∈ U alongthe set {h1, . . . , hn} and the mapsx → Df,x+ (hi),i = 1, . . . , n−1 are continuous atx0 (resp.f is directionally differentiableat every x∈Ualong wholeEand the mapsx→Df,x+ (hi),i= 1, . . . , n−1are continuous atx0).
PROOF. (a,a’)First we shall prove:
(a1) Proposition1(a,a’) is true whenFis a normed space.
Writeak:=Pn
j=khj,k= 1,2, . . . , n. It is needed to show that
t>0, t→0lim
∆f,x0,t(a1)−
n
X
k=1
D+f,x
0(hk)
= 0. (7)
Fixε >0. Let us find aδ >0so that
t>0, t→0lim
∆f,x0,t(a1)−
n
X
k=1
Df,x+
0(hk)
< δ, ∀t∈]0, δ[. (8) Since the mapsx→Df,x+ (hi),i= 1, . . . , n−1are continuous atx0, there is a neighborhoodV of zero inEsuch thatx0+V ⊂U and
kDf,x+ (hk)−D+f,x
0(hk)k< ε/4(n−1), ∀x∈V, k= 1, . . . , n−1. (9)
LetV0be a balanced neighborhood of zero inEsuch thatV0+V0⊂V. SinceV0is absorbing, there is a δ1>0such that
tak ∈V0, and thk∈V0 ∀t∈[0, δ1], k= 1,2, . . . , n.
Sincefis directionally right differentiable atx0alonghn, there is a0< δ2<1such that
∆f,x0,t(hn)−Df,x+
0(hn)
< ε/4 ∀t∈]0, δ2]. (10) Writeδ:= min(δ1, δ2)and fix a numbert∈]0, δ].
Since
f(x0+ta1)−f(x0) =
n−1
X
k=1
[f(x0+tak)−f(x0+tak+1)] + [f(x0+tan)−f(x0)],
we have
f(x0+ta1)−f(x0)
t −
n
X
k=1
D+f,x
0(hk) =
n−1
X
k=1
f(x0+tak)−f(x0+tak+1)−D+f,x
0+tak+1(thk) t
+
n−1
X
k=1
h D+f,x
0+tak+1(hk)−Df,x+
0(hk)] + [∆f,x0,t(hn)−D+f,x
0(hn)i .
SinceV0is balanced, for a fixedk < nwe have
x0+tak+sthk ∈x0+V0+V0⊂x0+V ⊂U, ∀s∈[0,1],
an application of Lemma 3 gives the existence of numberssk∈]0,1[such that kf(x0+tak)−f(x0+tak+1)−Df,x+
0+tak+1(thk)k
≤ kD+f,x
0+tak+1+skthk(thk)−D+f,x
0++tak+1(thk)k, k= 1, . . . n−1.
From this and preceding relation we obtain:
∆f,x0,t(a1)−
n
X
k=1
D+f,x
0(hk) ≤
n−1
X
k=1
D+f,x
0+tak+1+skthk(hk)−D+f,x
0+tak+1(hk)
+
n−1
X
k=1
Df,x+
0+tak+1(hk)−Df,x+
0(hk) +
∆f,x0,t(hn)−D+f,x
0(hn) .
Sincetak+1+skthk ∈ V,k = 1, . . . n−1andtak+1 ∈ V,k = 1, . . . n−1from (9) and triangle inequality we can write:
n−1
X
k=1
D+f,x
0+tak+1+skthk(hk)−Df,x+
0+tak+1(hk) < ε/2
Again from (9) we have:
n−1
X
k=1
Df,x+
0+tak+1(hk)−Df,x+
0(hk) < ε/4
From the last two relations and (10) we get:
∆f,x0,t(a1)−
n
X
k=1
D+f,x
0(hk)
< ε, ∀t∈]0, δ].
Sinceε >0is arbitrary, (8) (and hence(a1)too), is proved.
Let nowFbe an arbitrary Hausdorff locally convex space. Then there exists a family(Yj)j∈Jof normed spaces and a family(uj)j∈J of continuous linear mappingsuj:F → Yj, such that the topology ofF is determined by(uj)j∈J. Taking into account this fact and observation (S6), forFProp. 1(a)will be proved if we can show:
(a2)For a normed spaceY and for a continuous linear mappingu:F →Y
t>0, t→0lim
u
∆f,x0,tXn
k=1
hk
−
n
X
k=1
D+f,x
0(hk) Y
= 0.
Clearly,(a2)follows from the already proved(a1), applied to the mappingu◦f:U →Y.
(b, b0)Fix a continuous linear functionalu:F →R. An application of(a)to the functionu◦f :E→R gives thatu◦fis directionally right differentiable (is directionally differentiable) alongh1+· · ·+hnand
Du◦f,x+
0(h1+· · ·+hn) =D+u◦f,x
0(h1) +· · ·+Du◦f,x+
0(hn). (11)
Since, by assumption, f itself is directionally right differentiable (is directionally differentiable) along h1+· · ·+hn, from (11) we get:
u(Df,x+
0(h1+· · ·+hn)) =u(D+f,x
0(h1) +· · ·+D+f,x
0(hn)). (12)
Since (12) holds for an arbitrary continuous linear functionaluandF is dually separated, we get that (6) holds as well.
To simplify formulations, for an elementhand a nonempty subsetW0of an additive topological Abelian groupEwe writeh−W0:={h−x:x∈W0}and callW0an AO-set with respect toV ⊂Eif
W0∩(h1−W0)∩(h2−W0)6=∅ ∀h1, h2∈V. (13) Proposition 2 Let E be a real Hausdorff topological vector space, U ⊂ E be an open set, F be a Hausdorff topological vector space,f:U → F be a continuous mapping andW0 ⊂Ebe a AO-set with respect to a neighborhoodV of zero inE.
(a) IfF is locally convex,f is directionally right differentiable at everyx∈U alongW0and for every h∈W0the mapx7→D+f,x(h)is continuous atx=x0∈U, thenf is directionally differentiable at x0and the directional derivativeDf,x0:E→Fis a linear mapping.
(b) IfFis a (not necessarily locally convex) dually separated space,fis directionally right differentiable at everyx ∈ U along wholeE and for everyh ∈ W0 the map x → D+f,x(h)is continuous at x=x0∈U, then the directional derivativeDf,x0:E→Fis a linear mapping.
(a’,b’) The conclusion of (a) (resp. of (b)) remains true for a not necessarily continuousf:U →Fprovided f isdirectionally differentiable at everyx ∈ U alongW0 and for every h ∈ W0 the mapx 7→
Df,x+ (h)is continuous atx=x0 ∈U(resp. f isdirectionally differentiableat everyx∈U along wholeEand for everyh∈W0the mapx→Df,x+ (h)is continuous atx=x0∈U).
(c) In (a) and (b) in general it may happen that the linear mappingDf,x0:E→F is not continuous.
PROOF. Fix a neighborhoodV of zero inEfor which (13) holds.
Item(a,a’). Thanks to observations (S1) and (S5), the directional differentiability off atx0and the linearity ofDf,x0:E→Fwill be proved if we can show the validity of the next statement:
(V) For every h1, h2 ∈ V the mapping f is directionally right differentiable at x0 along the set {h1, h2, h1+h2}andD+f,x
0(h1+h2) =D+f,x
0(h1) +D+f,x
0(h2).
Leth1,h2∈V. Then, sinceW0is an AO-set with respect toV, we have:W0∩(h1−W0)∩(h2−W0)6=
∅. The last relation implies that there are elementsh0, h00, h000∈W0withh1=h000+h0, h2=h000+h0. Hence,h1+h2= 2h000+h0+h00.
Sincefis directionally right differentiable atx0along the set{h0, h00, h000}and the mapsx7→D+f,x(h0), x7→D+f,x(h00),x7→D+f,x(h000)are continuous atx0, by Proposition 1(a)we conclude:
(1)fis directionally right differentiable atx0along the vectorh000+h0=h1and D+f,x
0(h1) =Df,x+
0(h000+h0) =Df,x+
0(h000) +D+f,x
0(h0).
(2)fis directionally right differentiable atx0along the vectorh000+h00=h2and D+f,x
0(h2) =D+f,x
0(h000+h00) =Df,x+
0(h000) +D+f,x
0(h00).
(3)fis directionally right differentiable atx0along the vector2h000+h0+h00=h1+h2and D+f,x
0(h1+h2) =D+f,x
0(2h000+h0+h00) =D+f,x
0(2h000) +Df,x+
0(h0) +D+f,x
0(h00).
Clearly, from (1), (2) and (3) (sinceDf,x+
0(2h000) = 2Df,x+
0(h000)) we have also thatDf,x+
0(h1+h2) = D+f,x
0(h1) +D+f,x
0(h2)and (V) is proved.
Item(b,b’). In this case the proof is analogous, the only difference is that instead of Proposition 1(a), it is needed to use Proposition 1(b).
Item(c). It remains to show that in either of cases the directional derivativeDf,x0:E→Fmay not be continuous. This is shown in [6] by means of an example ofa uniformly continuousreal valued functional fgiven on a countably dimensional real inner product spaceE, with properties:
— f is directionally differentiable at everyx∈E,
— for everyh∈Ethe mappingx7→Df,x(h)is continuous onE,
— for everyx∈Ethe directional derivativeDf,x:E→Ris adiscontinuous linearfunctional.
Proposition 2 is applicable, e.g., whenW0=E. Some versions of Proposition 2(a’)forW0=Eand F=Rare contained in [20, Th. 24] and [29, Th. 2.1].
Lemma 4 LetEbe a Baire topological group (written additively),V an open neighborhood of zero inE, W0⊂V a set which is comeager inV. ThenW0is an AO-set with respect toV.
PROOF. Fix h1, h2 ∈ V and denoteV3 = V ∩(h1−V)∩(h2−V). Then0 ∈ V3, hence, V3 is a non-empty open set. SinceW0is comeager inV, we have thath1−W0is comeager inh1−V andh2−W0 is comeager inh2−V. Consequently,W0∩V3,(h1−W0)∩V3and(h2−W0)∩V3are comeager subsets of the non-empty open setV3, which is a Baire space as well. This implies that
W0∩(h1−W0)∩(h2−W0)∩V36=∅, hence, (13) is satisfied too.
The proof of Theorem 1 we need two more results, the first of which is a version ofthe Baire theorem.
Proposition 3 Let X,F be metrizable topological spaces, pn: X → F,n = 1,2, . . . be continuous mappings, p: X → F a mapping such that p(x) = limnpn(x),∀x ∈ X. Then the set C(p)of all continuity points ofpis comeager inX.
For a proof see [21,§31.X, Remark 5 and§34. VII].
The second needed result isthe Kuratowski-Ulam theoremobtained in [22], (see also [21,§22.V, Th. 1], [25, p. 56, Th. 15.1] and [23, Lemma 2.2]):
Proposition 4 LetOandY be separable metric spaces andW be a comeager subset ofO×Y. Then there exists a comeager subsetC⊂O, such that for everyx∈Cthe set{y∈Y : (x, y)∈W}is comeager inY.
Let us underline that the separability and the metrizability ofEin the proof of Theorem 1 will be used only through Proposition 4.
PROOF OFTHEOREM1
Case(a). Fix an open neighborhoodV0of zero inEsuch thatHis comeager inV0. First we will prove the next assertion:
(AA)If the assumptions of Theorem 1 are satisfied for an open setU having the formU = O+V, whereOis a non-empty open subset ofEandV ⊂V0is a balanced open neighbourhood of zero inE, then there exists a setR⊂O, comeager inO, such that for everyx∈Rthe functionfis Gˆateaux differentiable atx.
Proof of (AA). LetY =H∩V. SinceH is comeager inV0andV is open, we get thatY is comeager inV. In particular,Y 6=∅.
Then sincef is directionally right differentiable at everyx∈O⊂Ualongh∈Y ⊂H, we have:
(x, h)7→p(x, h) :=Df,x+ (h) = lim
n ∆f,x,1/n(h), ∀(x, h)∈O×Y.
SinceV is balanced we have thatx+ (1/n)·h ∈ U,∀x ∈ O,h ∈ V,n ∈ N. The last relation and the continuity off onU imply that the maps(x, h) 7→ ∆f,x,1/n(h),n = 1,2, . . . are continuous from O×Y toF; consequently, since the spacesO×Y andF are metrizable, we can apply to the mapping p:O×Y →F Proposition 3 and get that the setW := C(p)of all continuity points ofpis comeager in O×Y. Now, sinceEis a separable metrizable space, by Proposition 4 there exists a setCcomeager inO, such that
(KU) for everyx∈Cthe setWx={h∈Y : (x, h)∈W}is comeager inY.
SinceOis a Baire space, its comeager subsetCcontains a denseGδ-subset ofO; in particularC6=∅.
Now we fixx0∈C. It is sufficient to prove that:
(A)The mapfis directionally differentiable atx0andDf,x0:E →Fis linear.
(C)Df,x0:E→Fis continuous.
Proof of (A). DenoteW0=Wx0. From (KU) we have thatW0is comeager inY.SinceY is comeager inV, we obtain thatW0is comeager inV. From Lemma 4 we conclude thatW0is an AO-set with respect toV and write:
(1)W0∩(h1−W0)∩(h2−W0)6=∅, ∀h1, h2∈V.
By assumption,f is directionally right differentiable at everyx∈U along the setH. From this, since W0⊂H and andx0∈C, we obtain:
(2)fis directionally right differentiable at everyx∈Ualong the setW0and for everyh∈W0the map x7→D+f,x(h)is continuous atx0.
Taking into account (1) and (2), we can apply Proposition 2(a)and conclude that (A) is true.
Proof of (C). According to (A) for each givenh∈Ethe functionf is directionally differentiable atx0
alongh;hence,
Df,x0(h) = lim
n ∆f,x0,1/n(h), ∀h∈E. (14)
Since h 7→ ∆f,x0,1/n(h), n = 1,2, . . . are continuous mappings from a metrizable spaceV to a metrizable spaceF, it follows from (14) and Proposition 3 that the set C(Df,x0) ⊂ V of all continuity points ofDf,x0:V →F is comeager inV. SinceV is a Baire space, C(Df,x0)is not empty. From this
and the linearity ofDf,x0: E → F (which we have again by (A)) it follows that D+f,x
0: E → F is a continuous linear mapping. Thereforef is Gˆateaux differentiable atx0and (AA)is proved.
Now we show that Theorem 1 for the general case whenU 6=Eis an arbitrary open set can be derived from(AA).
Let(Vn)n∈Nbe a fundamental sequence of balanced open neighbourhoods of zero inEsuch thatVn+ Vn ⊂ Vn−1 ⊂ V0,n = 2,3, . . .. For each natural numberndenote byOn the complement inE of the closure of(E\U) +Vn. ThenOn⊂On+1⊂U,n= 1,2, . . . andU =∪∞n=1On. Fix a natural numberk so thatOk6=∅. Then we have alsoOn+Vn⊂On+2⊂U,n=k, k+ 1, . . .. Fix a natural numbern≥k.
We can apply(AA)to functionf:On+Vn →Fand get the existence of a setRn⊂On, comeager inOn, such that for everyx∈Rn the functionf is Gˆateaux differentiable atx. Then the setC =∪∞n=kRnwill be comeager inU and will have the property that for everyx∈Cthe functionf is Gˆateaux differentiable atx.
Case(b). In this case the proof is analogous, the only difference is that instead of Proposition 2(a), it is needed to use Proposition 2(b).
Remark 5 In case whenE is a complete metrizable space, in the above given proof the continuity of the directional derivative can be established also by means of well-known Banach’s theorem (see[5, Ch. I, Th. 4]).
Acknowledgement. The authors are grateful to Igor Orlov for drawing their attention on (the continu- ous version of) Lemma 1(a); to George Chelidze, Omar Dzagnidze, Elena Martin-Peinador and Ricardo Vidal for useful discussions.
The third author was partially supported by MCYT BFM2003-05878
Our special thanks go to the referee, whose critical remarks we have attempted to take into account maximally.
References
[1] Alexiewicz, A. and Orlicz, W. (1952). On the differentials in Banach spaces,Ann. Soc. Polon. Math.25, 95–99.
[2] Aronszajn, N. (1976). Differentiability of Lipschitzian mappings between Banach spaces,Stidia Math.57, 147–
190.
[3] Asplund, E. (1968). Fr´echet differentiability of convex functions,Acta Math.,121, 31–47.
[4] Averbukh V. I. and Smolyanov, O. G. (1967). The theory of differentiation in linear topological spaces,Russ.
Math. Surv.,22, 6, 201–258.
[5] Banach, S. (1978).Th´eorie des op´erations lin´eaires, Chelsea Publishing Company, New York. (The second edi- tion, a reprint of the first edition of 1932).
[6] Banakh, T., Corbacho, E., Plichko, A. and Tarieladze, V. (2006). Automatic continuity and linearity of directional derivatives (to appear).
[7] Benyamini Y. and Lindenstrauss, J. (2000).Geometric nonlinear functional analysis, v.I, AMS Colloquium Pub- lications, v. 48; Providence, R.I.
[8] Bogachev, V. I. and Shkarin, S. A. (1988). Differentiable and Lipshitzian mappings of Banach spaces,Math.
Notes44, 5, 790–798.
[9] Borwein, J. M. and Fabi´an, M. (1993). On convex functions having points of Gˆateaux differentiability which are not points of Fr´echet differentiability,Can. J. Math.,46, 1121–1134.
[10] Bourbaki, N. (1974).Topologie g´en´erale, Ch. 5–10, Diffusion C.C.L.S. Paris.
[11] Bourbaki, N. (1974).Fonctions d’une variable r´eelle, Diffusion C.C.L.S. Paris.
[12] Cartan, H. (1978).C´alculo differencial, Ediciones Omega, Barcelona.
[13] Deville, R., Godefroy, G. and Zizler, V. (1993).Smoothness and renormings in Banach spaces, Pitman Mono- graphs, v. 64, Longman, London.
[14] Dieudonn´e, J. (1972).El´ements d’analyse, 1, Fondaments de l’analyse moderne, Gautier-Villars, Paris.´ [15] Dorogovtsev, A. Ya. (1993).Mathematical analysis, v.I, “Lybid’”, Kyiv (in Ukrainian).
[16] Fabi´an, M. (1981). Differentiability via one-sided directional derivatives,Proc. Amer. Math. Soc.,82, 495–500.
[17] Fabi´an, M. J. (1997).Gˆateaux differentiability of convex functions and topology: Weak Asplund spaces, Canadian Math. Soc. Series of Monographs and Advanced Texts, Wiley interscience, New York e.a.
[18] Fabi´an, M., Habala, P., H´ajek, P., Montesinos, V. S., Pelant, J. and Zizler, V. (2001).Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, v. 8, Springer, New York.
[19] Fabi´an, M. and Preiss, D. (1991). On intermediate differentiability of Lipschitz functions on certain Banach spaces,Trans. Amer. Math. Soc.113, 733–740.
[20] Gˆateaux, R. (1922). Sur diverses questions de calcul fonctionell,Bull. Soc. Math. France50, 1–37.
[21] Kuratowski, K. (1966).Topology, v. I, Academic Press, New York.
[22] Kuratowski, C. and Ulam, S. (1932). Quelques propri´et´es topologiques du produit combinatoire,Fund. Math., 19, 247–251.
[23] Lau, Ka Sing and Weil, C. E. (1978). Differentiability via directional derivatives,Proc. Amer. Math. Soc., 70, 11–17.
[24] Mazur, S. (1933). ¨Uber konvexe Mengen in linearen normieren R¨aumen,Studia Math.4, 70–84.
[25] Oxtoby, J. C. (1996).Measure and Category. Second Edition (fourth printing), Springer-Verlag, New York.
[26] Phelps, R. R. (1989). Convex Functions, Monotone Operators and Differentiability,Lecture Notes in Mathemat- ics,1364, 1–114.
[27] Pshenichnyi, B. N. (1969).Necessary conditions for an extremum, Nauka, Moscow (in Russian; Engish version:
Marcel Dekker, New York, 1971).
[28] Sukhomlinov, G. A. (1937). Analitic functionals,Bull. Moscow Univ. (A)1, 2, 1–19 (Russian).
[29] Va˘ınberg, M. M. (1952). On the differential and gradient of the functionals,Uspehi Mat. Nauk,7, 3, 55–102 (in Russian).
[30] Yamamuro, S. (1974). Differential calculus in topological linear spaces,Lecture Notes in Mathematics, 179, 1–179.
[31] Zhivkov, N. V. (1987). Generic Gˆateaux differentiability of directionally differentiable mappings,Rev. Roumaine Math. Pures Appl.32, 2, 179–188.
[32] Zorn, M. A. (1945). Characterization of analytic functions in Banach spaces,Annals of Math.,46, 585–593.
E. Corbacho A. Plichko V. Tarieladze
Universidad de Vigo, Politechnika Krakowska im. Universidad de Vigo,
Dep. de Matem´atica Aplicada. Tadeusza Ko´sciuszki. Dep. de Matem´atica Aplicada.
36200, Vigo 31-155 Krak´ow 36200 Vigo, Espa˜na and
Espa˜na Poland Muskhelishvili Institute of Compu-
tational Mathematics.
Tbilisi-0193, Georgia.