V-jacobian and V-co-jacobian for lipschitzian maps

The notions of $V$-Jacobian and $V$-co-Jacobian are introduced for locally Lipschitzian functions acting 
between arbitrary normed spaces $X$ and $Y$, where $V$ is a subspace of the dual space $Y^*$. The main results 
of this paper provide a characterization, calculus rules and also the computation of these Jacobians 
of piecewise smooth functions.


1.
Introduction. Consider a Lipschitzian map f acting between two normed spaces X and Y . Motivated by the case of smooth maps, the task of finding a "good" generalized Jacobian notion at a point p focuses on searching for a derivative-like object, namely, a nonempty set of linear operators that serves as a reasonable approximation of the function change near p, and possesses "applicable" calculi.
The notion of Clarke's generalized gradient ( [4]) is known to be a satisfactory approximation for a real-valued, Lipschitzian function f at p by a nonempty subset of L(X, R), which is the dual space X * . Similarly, when X and Y are of finite dimension, Clarke's generalized Jacobian was defined for Lipschitzian maps in [3], [4] as a subset of the space L(X, Y ). Rademacher's differentiability theorem is instrumental in obtaining the nonemptiness of this generalized Jacobian.
For general normed spaces X and Y , several notions of sets playing the role of a derivative have been constructed. For instance, the notion of derivate containers in [31], [32]; the concepts of screens and fans in [6], [5]; the concept of shields [28]; the fan derivative [7], and the notion of coderivatives in [17]. Many of these notions are not given in terms of sets of linear operators, but rather in the form of set-valued maps.
On the other hand, in a series of papers [20], [19], [21], [22] and [23], notions of a generalized Jacobian and a co-Jacobian for Lipschitzian functions have recently 624 ZSOLT PÁLES AND VERA ZEIDAN been developed as nonempty sets of linear operators when X is any normed space. The notions are recalled in Section 3.
When Y is finite dimensional, the generalized Jacobian ∂f (p) ⊆ L(X, Y ) was constructed in [20] and [21] so that it extended both Clarke's generalized Jacobian and gradient. Moreover, the following basic identity was established: This identity relates Clarke's gradient to our generalized Jacobian and could be viewed as a special case of the chain rule. Subsequently, this notion of generalized Jacobian was expanded in [19] to the case when, for some normed space V , we have Y = V * and Y has the Radon-Nikodým property. In this more general setting, ∂f (p) is kept in the space L(X, Y ) (which equals L(X, V * )) and satisfies the linear chain rule (1.1). The Radon-Nikodým property of Y guarantees the applicability of the generalized Rademacher differentiability theorem, while the range being a dual space ensures that the norm-closed unit ball of the space L(X, Y ) is compact in a certain topology. A consequence of these two properties is that ∂f (p) is a nonempty subset of L(X, Y ). Therefore, when neither the range is a dual space nor the Radon-Nikodým property is present, one can naturally ask whether a concept of a generalized Jacobian ∂f (p) ⊆ L(X, Y ) can be defined in such a way that it is nonempty, and enjoys a sufficiently rich and applicable calculus, in particular, identity (1.1) is satisfied. In fact, we will show in an example below that such a goal cannot be reached without imposing extra assumptions on the range space Y .
In order to avoid assuming any condition on the range space, we introduced in [23] the notion of a co-Jacobian, ∂ * f (p), which is a nonempty set of operators in the space L(Y * , X * ), as opposed to L(X, Y ). This set ∂ * f (p) satisfies the following linear chain rule: which is a counterpart of (1.1). Furthermore, when Y = V * and Y has the Radon-Nikodým property, it is shown that When Y = V * , but Y does not have the Radon-Nikodým property, then the first equation in (1.3) motivates a definition for the generalized Jacobian that is still contained in L(X, Y ) and naturally generalizes the concept known for the Radon-Nikodým setting. It follows that when Y = V * then, V ⊆ Y * . Consider the case when Y is not the dual of a normed space V but only V ⊆ Y * holds. Then, the first equation in (1.3) is still capable of defining a generalized Jacobian. However, in this case, the set of operators defining ∂f (p) will no longer stay in L(X, Y ), but rather in L(X, V * ). Observe that from Lemma 2.2 we know that V ⊆ Y * yields that V * = Y * * | V ⊇ Y | V . If V is also separating the elements of Y , then the same lemma implies that Y | V isomorphic to Y and hence, in this case, we could consider the space V * as an enlargement of Y . The main advantage of this enlargement lies in the fact that Y is embedded into a dual space which brings us back to the previous friendly setting. It appears that a natural candidate for the space V is the largest possible choice, which is V = Y * . However, in this case, V * = Y * * , which can be significantly larger than Y when Y is non-reflexive. Thus, in the non-reflexive case, it is more rewarding to choose V as an appropriate proper subspace of Y * so that the gap between Y * * | V and Y | V is not too big.
From the first glance, it may appear that finding a generalized Jacobian for f : X → Y in the space L(X, V * ) instead of the space L(X, Y ) is somewhat unnatural and unreasonable. However, as we shall see in the simple example below, even if a minimal requirement is postulated for a concept of a generalized Jacobian, then this goal is not in general reachable.
Given any Lipschitzian map f : X → Y , we require that a generalized Jacobian ∆f : X → 2 L(X,Y ) must satisfy the following natural assumptions: (i) ∆f (x) is a nonempty subset of L(X, Y ) for all x ∈ X; (ii) For every linear functional y * ∈ Y * , the linear (inclusion) chain rule holds: In the context when X and Y are the spaces of continuous functions, we furnish an example which shows that the existence of a generalized Jacobian satisfying conditions (i) and (ii) is not possible. The reason for the failure of such a construction is that the gap between Y and Y * * is significant (see [9]). It is worth mentioning that the conclusion of this example remains valid if we replace in the above condition (ii) Clarke's gradient by any other smaller subdifferential.
Example. Denote by C(I) the space of continuous real-valued functions equipped with the supremum norm, where I := [−1, 1]. Define the function f : C(I) → C(I) by f (x)(t) := |x(t)|, and let x 0 (t) := t. We show that ∆f (x 0 ) ⊆ L C(I), C(I) cannot be defined so that properties (i), (ii) be satisfied.
Clearly f is a Lipschitzian map with Lipschitz modulus 1. For a fixed τ ∈ I, consider the linear functional y * τ ∈ (C(I)) * (which is the evaluation at the point τ ) defined by y * τ , y := y(τ ) (y ∈ C(I)).
Then y * τ • f is a Lipschitzian function and its Clarke's generalized directional derivative at x ∈ C(I) is computed in the following way: Therefore, a linear functional of C(I) represented by a bounded regular Borel mea- This is equivalent to the property that the support of the measure µ is the singleton {τ } and Hence µ({τ }) = sign(x(τ )) whenever x(τ ) = 0.
In other words, there is no reasonable form of a generalized Jacobian in the space L C(I), C(I) for all Lipschitzian maps f : C(I) → C(I). Therefore, the desired generalized Jacobian should be constructed in this case in another space, which is more appropriate. In fact, we shall show in the last section of this paper that such a Jacobian could be successfully constructed as a subset of L C(I), B(I) , where B(I) denotes the space of bounded real-valued functions equipped with the sup-norm. For this setting, the space V will be chosen as a proper subspace of Y * , namely, the linear subspace spanned by all the Dirac measures on the interval I.
The aim of this paper is to introduce the concepts of the V -co-Jacobian, ∂ * V f (p) ⊆ L(V, X * ), and the V -Jacobian, ∂ V f (p) ⊆ L(X, V * ) for a Lipschitzian map f acting between two normed spaces X and Y , where V is any subspace of Y * . More specifically, we define the set ∂ V f (p) as the largest subset of L(X, V * ) such that the following linear chain rule holds y y y * • ∂ V f (p) = ∂(y y y * • f )(p), (1.5) for all n ∈ N and for all y y y * ∈ V n . When n = 1 and V = Y * , this identity becomes the chain rule in (1.1). If (1.5) is required only for n = 1 then the set ∂ V f (p) would be much larger and hence less informative. In Sections 4 and 5 we establish the nonemptiness and a characterization of the V -co-Jacobian and the V -Jacobian, respectively. Furthermore, we derive an identity relating both notions. The results on the co-Jacobian obtained in [23] are employed. Differentiability properties and mean value theorems are obtained in Section 6. In Section 7 we derive the nonsmooth-smooth and the smooth-nonsmooth chain rules and hence, the sum rule. In Section 8 we establish the V -Jacobian, and hence V -co-Jacobian, for a continuous selection map. In the subsequent section we generalize Thibault's limit set and Ioffe's fan derivative so that they now take values in V * as opposed to Y . Furthermore, we establish the connections between our new notions and these modified concepts, as well as with Mordukhovich's normal and mixed co-derivatives.
2. Auxiliary results. Throughout this paper, whenever Z is a normed space, the symbols Z * and Z * * denote the first and second dual spaces of Z, respectively. The space Z is considered as a subset of Z * * via the canonical embedding. The open and closed unit balls of Z are denoted by B Z and B Z , respectively. The set Λ(Z) will denote the family of finite dimensional subspaces of Z.
Let X and Y be normed spaces and denote by L(X, Y ) the space of bounded linear operators from X to Y equipped with the standard operator norm. Given a continuous linear operator Φ : X → Y , its adjoint operator Φ * : Y * → X * is defined via the equality in other words, Φ * is given by the formula It is well-known that the adjoint operator Φ * is continuous linear and the map Φ → Φ * is a norm-preserving endomorphisms of L(X, Y ) into L(Y * , X * ). As we shall see below, these spaces are isometrically isomorphic whenever Y is reflexive.
To understand how the spaces L(X, Y ) and L(Y * , X * ) are related to each other when Y is a dual space of another normed space V , we need the following result.
Lemma 2.1. Let X and V be normed spaces. Then, for every Φ ∈ L(X, V * ), The mapping Φ → Φ * | V is an isometrical isomorphism from the space L(X, V * ) onto L(V, X * ) and its inverse is given by the mapping Ψ → Ψ * | X . Furthermore, and the identity mapping between these spaces is a linear isometry.
Indeed, using the definition of the adjoint operator twice, we get The linearity of the mapping Φ → Φ * | V is obvious. On the other hand, for every Φ ∈ L(X, V * ), we have which proves that Φ → Φ * | V is also an isometry. By the first part, we can see that the inverse of this mapping is given by Ψ → Ψ * | X (which is also a linear isometry). Hence both mappings are isometric isomorphisms.
To prove the first identity in Proof. By Lemma 2.1, we have that L(X, V * ) is isometrically isomorphic to the space L(V, X * ) = L(V * * , X * )| V , which proves that L(X, Y ) and L(Y * , , v * is a linear functional on V , then, by the Hahn-Banach extension theorem, v * can be extended to a linear functional Given two normed spaces X and V , we equip the space L(X, V * ) with a topology in which the norm-closed unit ball of L(X, The weak topology induced by X⊗V on L(X, V * ), i.e., the topology σ L(X, V * ), X⊗ V will be called the weak * -operator-topology and will be denoted by β(X, V ) throughout this paper. This notation indicates that the topology is described in terms of the elements of X and V . Obviously, the following sets form a neighborhood subbase for the origin in the β(X, V )-topology: that is, the β(X, V )-topology is the topology of the pointwise convergence for the real valued bilinear functions defined on X × V by (x, v) → Φ(x), v . Trivially, the β(X, V )-topology is weaker than the norm-topology, hence β(X, V )-closed sets are automatically norm-closed, and norm-compact sets are automatically β(X, V )compact.
The following theorem offers an analog of the Banach-Alaoglu theorem in the space L(X, V * ). Its proof can be found in [19, Thm. 2.1].
Theorem 2.4. ( [19]) Let X and V be arbitrary normed spaces. Then the normclosed unit ball of the Banach space L(X, V * ) is compact in the β(X, V )-topology. Remark 2.5. As a consequence of the above theorem, the norm-closed unit balls of the Banach spaces L(Y * , X * ) and L(X, Y * * ) are compact in the β(Y * , X)-and β(X, Y * )-topology, respectively. Hence, whenever Y is a dual of a normed space V , the norm-closed unit ball of L(X, Y ) is compact in the β(X, V )-topology.
Given arbitrary subspaces L ⊆ X and H ⊆ V , define the domain and image restriction maps dom L : L(X, V * ) → L(L, V * ) and im H : respectively. The following result follows immediately by applying the definition of the β topologies. It is also a consequence of [19, Proposition 2.2].
In order to prove inclusions between subsets of the space L(V, X * ), the following result will be needed. It follows from [19,Theorem 2.4]. Given an operator Φ ∈ L(X, It is obvious that f is an usc function on D and that f is Lipschitzian near p if and only if f (p) < +∞.
3.1. The Generalized Jacobian. For a real valued Lipschitzian function f : D → R, the generalized gradient of f at a point p ∈ D is defined by is Clarke's generalized directional derivative (see [4]). When X and Y are both finite dimensional normed spaces and f : D → Y is a vector-valued Lipschitzian function, Clarke introduced in [3], [4] the notion of the generalized Jacobian of f at a point p ∈ D by where Ω(f ) denotes the set of the points of D where f is differentiable. Based on Rademacher's celebrated differentiability theorem, we have that Ω(f ) is of full measure. In terms of the above generalized gradient and Jacobian, results have been derived pertaining optimality conditions, implicit functions theorems, metric regularity, and calculus rules including the sum rule and the chain rule. Thereby, it has already been shown that these objects are successful approximations of f by linear operators.
The notion of the generalized Jacobian introduced in [20] and [21] and which we recall here, will be the backbone for the co-Jacobian concept introduced in this paper. In those references, the difficulty caused by the infinite dimensionality of the domain was handled by introducing the following concepts of differentiability and Jacobian relative to finite dimensional subspaces of X, so that Rademacher's theorem remains applicable.
Clearly, the X-Gâteaux differentiability is equivalent to the standard Gâteaux differentiability. In this case, the subscript X will be omitted from the notation, i.e., D X f (p) will simply be denoted by Df (p). On the other hand, if L = h , that is, the linear span of a nonzero vector h ∈ X, then the L-Gâteaux differentiability of f means that the two-sided directional derivative, is Lipschitzian, therefore, by Rademacher's theorem, g is almost everywhere differentiable on L∩(D−p). Thus, there exist a sequence u i ∈ L such that u i → 0 and the sequence Dg(u i ) = D L f (p + u i ) converges. Based on this observation, we introduce the the L-Jacobian of f at p via the following formula: (3.6) Note that here the sequence (x i ) is not necessarily contained in the affine subspace p + L, and hence, ∂ L f (p) can be significantly larger than Clarke's generalized Jacobian of the restricted function f | p+L at p, which is ∂ c g(0), where g is the function defined above. We have that ∂ L f (p) is a nonempty compact convex set of the space L(L, Y ). Finally, we are able to recall the definition of the generalized Jacobian: If X is finite dimensional, then obviously ∂ X f (p) = ∂ c f (p). In this case, we have, for L ∈ Λ(X), that ∂ X f (p) L ⊆ ∂ L f (p). This yields that ∂ c f (p) ⊆ ∂f (p). Conversely, if Φ ∈ ∂f (p), then Φ = Φ| X ∈ ∂ X f (p) = ∂ c f (p), which implies the reversed inclusion ∂f (p) ⊆ ∂ c f (p). Therefore, the generalized Jacobian ∂f (·) extends Clarke's generalized Jacobian to the case when X is any normed space. If Y = R and X is any normed space, then ∂f (·) also coincides with Clarke's generalized gradient (cf. [21]). One of the main results established in the paper [21] is the theorem below that characterizes ∂f (·) as a smallest operator set-valued mapping satisfying certain properties. If the dimension of Y is N , then the space L(X, Y ) is topologically isomorphic to the product space (X * ) n , hence the space L(X, Y ) can be equipped with a the weak * topology inherited from (X * ) n .
Given a normed space Z equipped with a Hausdorff topology τ , a map F : D → 2 Z is said to be sequentially τ -usc at p ∈ D if, whenever (x i , z i ) is a sequence in D × Z such that z i ∈ F(x i ) for all i, and (x i ) tends to p, then τ -clus i→∞ z i ⊆ F(p), i.e., F(p) contains all the τ -cluster points of the sequence (z i ). (ii) F is sequentially w * -upper semicontinuous on D; (iii) For all L ∈ Λ(X) and for all x ∈ Ω L (f ), For further results, such as calculus rules, a mean value theorem, the computation rule of the generalized Jacobian of piecewise smooth functions, etc., we refer to the papers [20], [21].
In the case when Y is an infinite dimensional dual space with the Radon-Nikodým property, an involved generalization of the above construction was elaborated in [19] which led to an extension of the generalized Jacobian ∂f (·) to this setting. The underlying technique employed in [19] is the generalization of Rademacher's differentiability theorem proven in [2] and [1].
3.2. The Co-Jacobian. Based on the notion of the generalized Jacobian for functions with finite dimensional range, we have introduced in [23] the following notion of the co-Jacobian which will be useful in this paper.
Given y y y * = (y * 1 , . . . , y * n ) ∈ (Y * ) n , the finite dimensional-valued function g(x) := y y y * (called the vectorization of f ) is Lipschitzian near p provided that f is Lipschitzian near p. Thus, the notion of generalized Jacobian recalled in (3.7) can be now applied to the function g.
(3.11) and hence, for all n ∈ N and y y y * ∈ (Y * ) n , ∂ * f (p)(y y y * ) = ∂(y y y * • f )(p). (3.12) The following result from the paper [23] offers a complete characterization of the co-Jacobian as a set-valued map with certain properties. The result is analogous to Theorem 3.1. for all x ∈ D. (ii) F is sequentially β(Y * , X)-usc on D.

4.
Main results: V -co-Jacobian. Let X and Y be normed spaces, D be a nonempty open subset of X, p be an arbitrary point in D and f : D → Y be a Lipschitzian function. One of the main two notions of this paper is the V -co-Jacobian defined, for any subspace V of Y * , by ∂ * V f (p) := Ψ ∈ L(V, X * ) : ∀ n ∈ N, ∀ y y y * ∈ V n , Ψ(y y y * ) ∈ ∂(y y y * • f )(p) . Observe that this definition extends the notion defined in (3.10) for finite dimensional subspaces to any subspace of Y * . As it will be shown, it has an intimate relation to our new concept, the V -Jacobian ∂ V f (p), introduced in the next section.
The following result is an extension of Theorem 3.2 to the setting of arbitrary subspaces of V * . Its proof is based on the application of Theorem 3.2. ∂ It results that H∈Λ(V ) S H = ∅. That is, there exists Ψ ∈ ∂ * f (p) satisfying Ψ| V = Ψ 0 . Therefore, Ψ 0 ∈ ∂ * f (p)| V .
As an obvious consequence of Theorem 4.1, we obtain a result that describes the connection between V -co-Jacobians belonging to different subspaces of Y * .
The result below describes the essential properties of the V -co-Jacobian as a set-valued map. (iii) For all n ∈ N, y y y * ∈ V n , L ∈ Λ(X), and for all x ∈ Ω L (y y y * • f ), Then, by Lemma 2.6, Γ is a (β(Y * , X), β(V, X))-continuous linear operator with Γ 1 ≤ 1. Observe that, by Proposition 5.1, we have that . The function f being upper semicontinuous at x, it follows that f is bounded by f (x) + 1 in a neighborhood of x. Thus, the sequence (Ψ i ) is β(Y * , X)-precompact. Applying now [19,Lemma 2.7], it follows that On the other hand, by the β(Y * , X * )-usc property of the co-Jacobian map ∂ * f , we have that β(Y * , X * )-clus which proves the β(V, X)-usc property of the V -co-Jacobian map ∂ * V f at x. For proving (iii), let L ∈ Λ(X), n ∈ N, and y y y * ∈ V n such that y y y * • f is Ldifferentiable at x ∈ D. Then, by property (iii) of Theorem 3.3 and Theorem 4.1, we obtain Thus the proof of property (iii) is complete. Now assume that F V is a set-valued map satisfying the conditions (i)-(iii) of the theorem. Let p ∈ D be a fixed point where we want to show that the inclusion By property (i), co F V (p) is bounded, therefore the right hand side of (4.4) is β(V, X)-compact. Thus, in view of Theorem 2.7, it suffices to show that, for all n ∈ N, y y y * ∈ V n , and L ∈ Λ(X), By the linearity of the restriction and evaluation maps, we have co F V (p) (y y y * ) L = co F V (p)(y y y * ) L = co F V (p)(y y y * ) L .
Applying also the continuity properties of these maps established in Lemma 2.6 and Lemma 2.1, we obtain co β(V,X) F V (p) (y y y * ) L = co β(X,R n ) F V (p)(y y y * ) L = co F V (p)(y y y * ) L .
To simplify the left hand side of (4.7), let δ > 0 be such that (4.4) holds for all x ∈ Ω L (y y y * • f ) ∩ (p + δB X ). By Theorem 3.2, we have where ∂ L (y y y * • f )(p) is defined by Hence (4.7) is equivalent to proving Thus, in order to verify (4.8), it suffices to show that (4.9) Indeed, if Φ is an element of the left hand side then there exists a sequence (x i ) in Ω L (y y y * •f ) such that lim i→∞ x i = p and lim i→∞ D L (y y y * •f )(x i ) = Φ. By assumption (iii), holds for large i. Thus, applying property (ii), it follows that showing that (4.9) holds. The following theorem enlightens the connection between the V -Jacobian and the V -co-Jacobian of f . As a consequence, we obtain that the generalized Jacobian introduced in the papers [20], [21], and [19], is completely determined by the co-Jacobian.

Main results
Proposition 5.1. Let f : D → Y be a Lipschitzian function p ∈ D and V be a subspace of Y * . Then, Proof. Using (2.1), it easily follows from the definitions of ∂ V f (p) and  Then where ∂f (p) denotes the generalized Jacobian of f at p defined in [19].
Proof. By [19,Theorem 3.7], it follows that, for all n ∈ N and y y y * ∈ V n , y y y * • ∂f (p) = ∂(y y y * • f )(p), (iii) For L ∈ Λ(X), n ∈ N, and y y y * ∈ V n such that y y y * • f is L-differentiable at x ∈ D, we have D L (y y y * • f )(x) ∈ y y y * • F V (x) L . Observe that, by Proposition 5.1, we have that, for all x ∈ D, On the other hand, by Lemma 2.1, Γ is a (β(V, X), β(X, V ))-continuous isometric isomorphism between the spaces L(V, X * ) and L(X, V * ). Hence, the properties of the V -co-Jacobian map ∂ * V f =: F V presented in Theorem 4.3 directly yield that the V -Jacobian map ∂ V f =: F V enjoys the properties listed in (i)-(iii) of Theorem 5.5.
To prove the reversed statement, assume that F V : D → 2 L(X,V * ) is a set-valued map satisfying conditions. (i)-(iii). Then, it is easy to see that the set-valued map satisfies all conditions (i)-(iii) of Theorem 4.3. Therefore, we get, for all x ∈ D, that , which completes the proof. 6. Differentiability properties and mean value Theorem. In this section we describe how the V -Jacobian is connected to the differentiability properties of a Lipschitzian function f : D → Y at p ∈ D.
For a given subspace V ⊆ Y * , we introduce the notions of strict V -Fréchet, V -Hadamard, and V -Gâteaux prederivatives. A set of operators F ⊆ L(X, V * ) is called a strict V -Hadamard prederivative for the function f at p if, for all n ∈ N, for all y y y * ∈ V n , for all ε > 0, and for all compact subsets C of the unit sphere of X, there exists δ > 0 such that, for all x, y ∈ p + δB X with y − x ∈ y − x C, i.e., if y y y * • F ⊆ L(X, R n ) is a strict Hadamard-prederivative for the function y y y * • f at p for all y y y * ∈ V n . If the above requirements holds when C is the entire unit sphere, then F is called a strict V -Fréchet prederivative. On the other hand, when the above definition holds only for finite C, then F is called a strict V -Gâteaux prederivative. If F is a singleton, i.e., F = {Φ} and F is a strict V -Fréchet, V -Hadamard, or V -Gâteaux prederivative for f at p, then we say that f is strictly V -Fréchet, V -Hadamard, or V -Gâteaux differentiable at p with a V -derivative Φ, respectively and the V -derivative will be denoted by D V f (p).
Proof. By [23], we know that F := ∂ * f (p) is a strict w-Hadamard pre-coderivative for f at p. In other words, for all n ∈ N, for all y y y * ∈ (Y * ) n , for all ε > 0, and for all compact subsets C of the unit sphere of X, there exists δ > 0 such that, for all x, y ∈ p + δB X with y − x ∈ y − x C, In particular, taking y y y * ∈ V n and using (5.2), the statement follows.
As a corollary, we obtain a characterization of the case when ∂ V f (p) is a singleton. The result is analogous to what is known for Clarke's subgradient and for the generalized Jacobian introduced in [20], [20], [21]. Conversely, if f is strictly V -Hadamard differentiable at p, then, for all and y * ∈ V , the real-valued function y * • f is Hadamard differentiable at p and hence, by the properties of generalized Jacobian, ∂(y * • f )(p) is a singleton. Thus, in view of Theorem 3.2, y * • ∂ V f (p) is also a singleton, for all y * ∈ V . It easily follows that ∂ V f (p) must be a singleton, too.
The next result, which is phrased in terms of our co-Jacobian, is a counterpart of the mean value theorems in terms of the generalized gradient and Jacobian (see, e.g., [13], [19], [21]).
For an element y ∈ Y , we define y| V ∈ V * by the formula y| V (y * ) := y * , y . In particular, if V = Y * , then y| V = y, where y is considered to be an element of Y * * via the canonical embedding.
Proof. Let x, y ∈ D be fixed with [x, y] ⊆ D. By the mean value theorem established in [23], we have where, for x ∈ X and Ψ ∈ L(Y * , X * ), the functional x • Ψ ∈ Y * * is defined by x • Ψ(y * ) = Ψ(y * ), x . Restricting both sides to the subspace V , and applying Lemma 2.6, Lemma 2.1 and the linearity of the restriction map and the conjugation, we obtain that whence, by applying Proposition 5.1, the result follows.
Remark 6.4. The inclusion (6.9) in the mean value theorem can also be phrased in terms of the V -co-Jacobian as follows: 7. Chain rules and their consequences. In this section we shall establish a nonsmooth-smooth and a smooth-nonsmooth chain rule. The next theorem, which is an extension of [21,Thm. 4.2], is called the nonsmooth-smooth chain rule.
If g is strictly W -Hadamard differentiable at f (p), then g • f is W -Hadamard differentiable at p and (7.1) holds with equality.
Proof. By the nonsmooth-smooth chain rule in terms of the co-Jacobian of g at f (p) derived in [23], we have that Hence, which, by Proposition 5.1, results that If g is strictly W -Hadamard differentiable at f (p) then ∂ W g f (p) is a singleton and hence (7.1) yields that ∂ W (g • f )(p) is also a singleton and the equality in (7.1) automatically holds.
Remark 7.2. Using Theorem 4.1, the inclusion in (7.2) can be rewritten as , which states the nonsmooth-smooth chain rule in terms of the W -co-Jacobians of g and g • f .
The following result is our smooth-nonsmooth chain rule. Then, In particular, if V * = Y and W * = Z hold, then Proof. Using the notation Ψ := Dg f (p) , it follows from (7.3) that By the smooth-nonsmooth chain rule in terms of the co-Jacobian derived in [23], we have that Hence, Thus, applying Proposition 5.1, equation (7.7) implies that , which proves (7.4). If we also have V * = Y and W * = Z, then which states the nonsmooth-smooth chain rule in terms of the co-Jacobians of g and f .
Using Theorem 7.3, the following more general smooth-nonsmooth chain rule follows.
Corollary 7.5. Let Y 1 , . . . , Y k , and Z be normed spaces and let V 1 , . . . , V k and W be subspaces of Y * 1 , . . . , Y * k , and Z * , respectively. Let f 1 : D → Y 1 , . . . , f k : D → Y k be Lipschitzian functions near p ∈ D and let g : O → Z be strictly Fréchet differentiable is an open set containing the point q. For all j ∈ {1, . . . , k}, denote D j g(q)(y j ) := Dg(q)(0, . . . , y j , . . . , 0) and assume that W • D j g(q) ⊆ V j . Then where the linear operators Φ j ∈ L(W, V j ) are defined by If W * = Z and also V * j = Y j holds for all j ∈ {1, . . . , k}, then • ∂ Vj f j (p). (7.10) Proof. The statement follows when Theorem 7.3 is applied to the functions g and f = (f 1 , . . . , f k ) and the easy-to-obtain inclusion is used.
As a particular case of Corollary 7.5 we obtain the so-called sum rule.
Corollary 7.6. Let Y be a normed space and V be a subspace of Y * . Let f 1 , f 2 : D → Y be Lipschitzian functions near p ∈ D. Then If either f 1 or f 2 are strictly V -Hadamard differentiable at p, then (7.12) holds with equality.
Proof. We apply Corollary 7.5 with the spaces Y 1 := Y 2 := Z := Y and V 1 := V 2 := W := V to the function g : Y 1 × Y 2 → Z defined by g(y 1 , y 2 ) := y 1 + y 2 . Then, for the point q = (f 1 (p), f 2 (p)) ∈ Y 1 × Y 2 , we have that D 1 g(q) = D 1 g(q) = id Y , where id S denotes the identity function of a set S into itself. Therefore, for j = 1, 2, the equality W • D j g(q) = V j trivially holds. On the other hand, for all w ∈ W = V , we get Φ j (w) = w • D j g(q) = w. Hence, Φ j = id V and Φ * j = id V * . Thus, formula (7.8) reduces to the sum rule (7.12).
For the equality, assume, say, that f 2 is V -Hadamard differentiable at p. Then, by Corollary 6.2, we have that ∂ V f 2 (p) = {D V f 2 (p)}. Thus applying (7.12), we get showing the equality in (7.12).
In the next result, we deduce a certain product rule, which can be useful in the applications.
Lemma 7.7. Let X, Y 1 , Y 2 , and Z be normed spaces and assume that the operation : Y 1 × Y 2 → Z is a bounded bilinear function, i.e., there exists a constant C such that, for all (y 1 , y 2 ) ∈ Y 1 × Y 2 , the inequality y 1 y 2 ≤ C y 1 y 2 holds. Let f = (f 1 , f 2 ) : D → Y 1 ×Y 2 be a Lipschitzian function. Then the function F : D → Z defined by is strictly Fréchet differentiable at p and DF (p) = 0.
Proof. Let r > 0 be chosen such that f 1 and f 2 are Lipschitzian on p + rB X with Lipschitz modulus L 1 and L 2 , respectively. For x, y ∈ p+rB X , we have the following estimate: F (x) − F (y) x − y = 0, which proves that Φ = 0 is a strict Fréchet derivative for F at p.
Theorem 7.8. Let X, Y 1 , Y 2 and Z be normed spaces, W ⊆ Z * be a subspace and let f = (f 1 , f 2 ) : D → Y 1 × Y 2 be a Lipschitzian function. Assume that : Y 1 × Y 2 → Z is a bounded bilinear function. Then ∂ W (f 1 f 2 )(p) = ∂ W f 1 f 2 (p) + f 1 (p) f 2 (p). (7.14) Proof. Define the function F : D → Z by (7.13). Then, by Lemma 7.7, F is strictly W -Hadamard differentiable and its W -Hadamard derivative is zero at p. On the other hand, we have that f 1 f 2 = F + f 1 f 2 (p) + f 1 (p) f 2 . Therefore, the sum rule (in the case when equality holds) applies, and (7.14) follows.
8. V -Jacobian for continuous selections. The notion of piecewise smooth function is a function whose domain can be partitioned into finitely many "pieces" relative on which smoothness holds and continuity holds across the joins of the pieces. In this section we consider functions that are piecewise locally Lipschitzian. Given a finite system of some Lipschitzian functions g 1 , . . . , g k : D → Y , a continuous function f : D → Y is called a continuous selection of {g 1 , . . . , g k } if, for all x ∈ D, there exists an index j ∈ {1, . . . , k} such that f (x) = g j (x), that is, for all x ∈ D, f (x) ∈ {g 1 (x), . . . , g k (x)}.
The main result of this section is Theorem 8.1 below which offers an inclusion for the generalized V -Jacobian of a Lipschitzian function which is decomposed in terms of finitely many Lipschitzian functions. The proof is based the analogous result obtained in terms of the co-Jacobian in our previous paper [23]. Furthermore, if the functions g j , for j in I(p), are strictly V -Hadamard differentiable at p, then (8.1) holds with equality.
Proof. By the result obtained in the setting of the co-Jacobian in [23], we have that ∂ * f (p) ⊆ co j∈I(p) ∂ * g j (p) . Hence, applying Proposition 5.1 twice and using obvious set theoretical and linear identities, we get If the functions g j , for j in I(p), are strictly V -Hadamard differentiable at the point p then (8.3) holds with equality and, by the above argument, (8.1) is also satisfied with equality.
Remark 8.2. The inclusion in (8.1) can also be phrased in terms of the V -co-Jacobian as: It easily follows from the known properties of Clarke's subgradient that the set t∈T ∂ c ϕ(p(t)) is bounded, hence the function t → ψ t is bounded. Thus, we have shown the following result: Proposition 10.1. Under the notations and assumptions above, for every point p ∈ D and for every linear map Φ ∈ ∂ V f ϕ (p), there exists a bounded function ψ : T → X * 0 such that ψ(t) ∈ ∂ c ϕ(p(t)) (t ∈ T ) (10.4) and (Φ(x))(t) = ψ(t), x(t) (x ∈ X, t ∈ T ).