Definition A.1. Let X be a (real or complex) normed space and let V be a locally convex space over the same field whose topology is defined by the family
qλ:λ∈L of seminorms which separate the points ofV. Suppose further that we are given an open subsetGofX.
(a) A mapping F :G→Vis called differentiable at the point x∈G if there exists a continuous linear mapping T :X→V such that for any vector h in a certain 0-neighbourhoodU⊆Xthe increment F(x+h)−F(x)allows for the linearized approx-imation
F(x+h)−F(x) =Th+R[F,x](h), (A.1a) where R[F,x]is a mapping onUtoVsubject to the condition
h→0limkhk−1qλ R[F,x](h)
=0 (A.1b)
for any seminorm qλ,λ∈L. The linear operator T occurring in (A.1a) is signified as DF(x)and called the derivative of F atx.
(b) The mapping F :G→Vis called differentiable if it is differentiable at anyx∈G. (c) The differentiable mapping F :G→Vis called continuously differentiable if the mappingG3x7→DF(x)h∈V, which exists by assumption, is continuous with respect to the locally convex topology ofVfor any givenh∈X.
Remark. The definition of the continuous linear operator DF(x) requires uniqueness of the corresponding T in (A.1a), but this is easily established. Assume the existence of another0-neighbourhoodU0, a continuous linear operator T0 :X→Vand a mapping
102 Concepts of Differentiability
R0[F,x]:U0→Vwhich, upon insertion into (A.1a), represent the increment F(x+h)− F(x)such that R0[F,x]fulfills a condition analogous to (A.1b). Then
Th−T0h=R0[F,x](h)−R[F,x](h), h∈U∩U0.
Lety∈X, y6=0, be arbitrary but fixed, then forα∈C\ {0}small enough we infer from the above equation due to the linearity of both T and T0
qλ Ty−T0y
=qλ α−1 R0[F,x](αy)−R[F,x](αy)
=kyk kαyk−1qλ R0[F,x](αy)−R[F,x](αy) , where the right-hand side vanishes in the limitα→0 for any seminorm qλ, according to (A.1b). This yields qλ Ty
=qλ T0y
, valid also fory=0, and as a consequence Ty=T0yfor anyy∈Xsince the seminorms qλseparate the points inV.
An immediate consequence of the presumed continuity of the linear operatorsDF(x), entering as derivatives the representation (A.1a) of the increment of F atx, is the fact that differentiability implies continuity.
Corollary A.2. LetXbe a normed space and letVbe a locally convex space. If the mapping F :G→V, G⊆Xopen, is differentiable at the pointx∈Gthen it is also continuous inx.
The methods used in the standard theory of differentiable functions yield the follow-ing propositions when applied to the concept laid open in Definition A.1, the main modification being the occurrence of seminorms qλonVin (A.1b).
Proposition A.3 (Product Rule for Derivatives). LetXbe a normed space andGan open subset ofX.
(i) Suppose thatVis a locally convex space and that the mappings F :G→Vand f :G→K,Kthe scalar field of bothXandV, are differentiable atx∈G. Then their product f F is differentiable at this point, too, and the derivative atxis given by
D(f F)(x)h=Df(x)hF(x) +f(x)DF(x)h, h∈X.
(ii) LetYbe a normed algebra and assume that the mappings F :G→Yand G :G→ Yare differentiable atx∈G. Then their product FG is differentiable atx, too, and the derivative is
D(FG)(x)h=DF(x)hG(x) +F(x)DG(x)h, h∈X.
Proposition A.4 (Chain Rule for Derivatives). LetXandY be normed spaces and let V be a locally convex space. Assume further that the mapping G :G1→ Y is differentiable atx∈G1 and that the mapping F :G2→V is differentiable at G(x), whereG1andG2are open subsets ofXandY, respectively, and G(G1)⊆G2. Then the composition of F and G: F◦G :G1→V, exists and is differentiable at xwith a derivative connected to those of F and G through
D(F◦G)(x) =DG F(x)
◦DF(x).
A.1 Differentiation in Locally Convex Spaces 103
The fundamental Mean Value Theorem which has to be formulated in the setting of DefinitionA.1is based on the following two lemmas. Their proof as well as that of the theorem proper is an adaptation of the reasoning to be found in [37, Kapitel XX, Abschnitt 175].
Lemma A.5. Let F :[a,b]→V be a continuous mapping on the compact interval [a,b]⊆Rto the locally convex space V and suppose that it is differentiable on the interior of this set withDF(x) =0 for any x∈]a,b[. Then F is constant on[a,b].
Proof. Let s and t be arbitrary distinct points in]a,b[. We shall assume s<t and want to show that F(s) =F(t). Define u .
=2−1(t−s)and consider one of the seminorms qλ topologizingV. There are two possibilities:
qλ F(u)−F(s)
>qλ F(t)−F(u)
, (A.2a)
qλ F(t)−F(u)
>qλ F(u)−F(s)
. (A.2b)
Depending on the actual situation we define an interval]s1,t1[⊆[a,b], choosing s1 .
=s, t1 .
=u in case (A.2a) and s1 .
=u, t1 .
=t in case (A.2b). Independent of this selection is the estimate
qλ F(t)−F(s)
6qλ F(t)−F(u)
+qλ F(u)−F(s)
62 qλ F(t1)−F(s1) . (A.3) The same procedure can then be applied to the interval]s1,t1[, to the resulting interval ]s2,t2[and so on. In this way a sequence of intervals ]sn,tn[is constructed, which is decreasing with respect to the inclusion relation: ]sn+1,tn+1[⊆]sn,tn[. Furthermore the lengths are explicitly known as tn−sn=2−n(t−s)and the estimate (A.3) can be generalized to
qλ F(t)−F(s)
62nqλ F(tn)−F(sn)
. (A.4)
There exists exactly one point u0 ∈]a,b[belonging to all intervals of this sequence and by assumptionDF(u0) =0, so that for h in a small 0-neighbourhood U⊆Rthe increment of F at u0is represented by
F(u0+h)−F(u0) =h R(h) (A.5a) with a mapping R :U→Vsatisfying
h→0limqλ R(h)
=0. (A.5b)
Hence, givenε>0, there exists N∈Nsuch that qλ R(u0−sn)
and qλ R(tn−u0) are majorized by(t−s)−1εfor n>N. According to (A.5a) this implies
qλ F(tn)−F(sn)
6qλ F(tn)−F(u0)
+qλ F(sn)−F(u0) 6|tn−u0|qλ R(tn−u0)
+|u0−sn|qλ R(u0−sn) 6(tn−u0) ε
t−s+ (u0−sn) ε
t−s = (tn−sn) ε t−s = ε
2n,
104 Concepts of Differentiability
where we made use of the length formula for the interval]sn,tn[. From (A.4) one then infers
qλ F(t)−F(s) 62n ε
2n =ε,
so that, by arbitrariness ofεand qλtogether with the separation property of the semi-norms, we see that F(t) =F(s) =v0∈V. This relation holds for any s, t ∈]a,b[
and extends by the supposed continuity of F to all of [a,b], establishing F≡v0 as stated.
Lemma A.6. Let F :[a,b]→V be a continuous mapping on the compact interval [a,b]⊆Rto the locally convex spaceVand define G :[a,b]→V,Vthe completion of V, through the integral
G(x) .
= Z x
a
dϑF(ϑ), x∈[a,b].
Then the mapping G is differentiable for any x0∈]a,b[and the action of the derivative DG(x0)as a linear operator onRis given by
DG(x0)h=h F(x0), h∈R. (A.6) Proof. By [26, II.6.2] G is a well-definedV-valued mapping on the compact interval [a,b]. For x0∈]a,b[and h∈Rsatisfying x0+h∈[a,b]we have
G(x0+h)−G(x0) = Z x0+h
x0
dϑF(ϑ),
hence
G(x0+h)−G(x0)−h F(x0) = Z x0+h
x0
dϑ F(ϑ)−F(x0) .
=ρ(h). (A.7a) Now by assumption,ϑ7→qλ F(ϑ)−F(x0)
is continuous on the compact interval Ih
of integration for any of the defining seminorms qλofV, and, according to [26, II.6.2 in connection II.5.4], one has for any h the estimate
|h|−1qλ ρ(h)
6|h|−1
Z x0+h x0
dϑqλ F(ϑ)−F(x0) 6max
ϑ∈Ih
qλ F(ϑ)−F(x0) , (A.7b) where the right-hand side vanishes in the limit h→0. Thus (A.7a) corresponds to the representation (A.1a) of Definition A.1in terms of the increment G(x0+h)−G(x0) with a residual termρ(h)satisfying (A.1b). This proves differentiability of G on]a,b[
along with relation (A.6).