Dual and Generalized Dual Cones in Banach Spaces

The primary objective of this paper is to propose and analyze the notion of dual cones associated with the metric projection and generalized projection in Banach spaces. We show that the dual cones, related to the metric projection and generalized metric projection, lose many important properties in transitioning from Hilbert spaces to Banach spaces. We also propose and analyze the notions of faces and visions in Banach spaces and relate them to the metric projection and generalized projection. We provide many illustrative examples to give insight into the given results.

The primary objective of this research is to propose and analyze the notion of dual cones associated with the metric projection in Banach spaces.We note that the shortcomings of the metric projection in Banach spaces have resulted in important extensions, namely, the generalized projection and the generalized metric projection, which enjoy better properties in a Banach space framework, see [33,34,35,22,36].We show that the dual cones, related to the metric projection and generalized metric projection, lose many properties in transitioning from Hilbert spaces to Banach spaces.We also propose and analyze the notions of faces and visions in Banach spaces and relate them to the metric projection and generalized projection.Illustrative examples are given.
The contents of this paper are organized into five sections.After a brief introduction in Section 1, we recall some background material Section 2. Section 3 studies dual cones related to the metric projection where as the dual cones related to the generalized projection are studied in Section 4. various notions of projections and give new results concerning normalized duality mapping.Section 5 studies the faces and visions in Banach spaces.

Preliminaries
Let X be a real Banach space with norm • X , let X * be the topological dual of X with norm • X * , and let •, • be the duality pairing between X * and X.We will denote the null elements in X and X * by θ and θ * .Moreover, the closed and convex hull of a set M ∈ X is denoted by co(M).Given a Banach space X, for r > 0, we denote the closed ball, open ball and sphere with radius r and center θ by For details on the notions recalled in this section, see [37].Given a uniformly convex and uniformly smooth Banach space X with dual space X * , the normalized duality map J : X → X * is a single-valued mapping defined by In a uniformly convex and uniformly smooth Banach space X, the normalized map J X : X → X * is one-to-one, onto, continuous and homogeneous.Furthermore, the normalized duality mapping J * : X * → X is the inverse of J, that is J * J = I X and JJ * = I X * , where I X and I * X are the identity maps in X and X * .On the other hand, in a general Banach space X with dual X * , the normalized duality mapping J : X → 2 X * is a set-valued mapping with nonempty valued.In particular, if X * is strictly convex, then J : X → X * is a single-valued mapping.See [37].
The following example will be repeatedly used in this work.
Example 2.1.Let X = R 3 be equipped with the 3-norm • 3 defined for any z = (z 1 , z 2 , z 3 ) ∈ X, by Then (X, • 3 ) is a uniformly convex and uniformly smooth Banach space (and is not a Hilbert space).The dual space of (X, ) so that for any ψ = (ψ 1 , ψ 2 , ψ 2 ), we have The normalized duality mappings J and J * satisfy the following conditions.For any z = (z 1 , z 2 , z 3 ) ∈ X with z = θ , we have Moreover, for any ψ = (ψ 1 , ψ 2 , ψ 3 ) ∈ X * with ψ = θ , we have Let X be a uniformly convex and uniformly smooth Banach space and let C be a nonempty, closed, and convex subset of X.We define a Lyapunov function V : X * × X → R by the formula: We shall now recall useful notions of projections in Banach spaces.
Definition 2.2.Let X be a uniformly convex and uniformly smooth Banach space, let X * be the dual of X, and let C be a nonempty, closed, and convex subset of X.

The generalized projection π
The following result collects some of the basic properties of the metric projection defined above.
Proposition 2.3.Let X be a uniformly convex and uniformly smooth Banach space and let C be a nonempty, closed, and convex subset of X.
1.The metric projection P C : X → C is a continuous map that enjoys the following variational characterization: 2. The generalized projection π C : X * → C enjoys the following variational characterization: For any ψ ∈ X * and y ∈ C, y = π C (ψ), if and only if, ψ − J X y, y − z ≥ 0, for all z ∈ C. (5) We will also need the following notions.Given a Banach space X, for any u, v ∈ X with u = v, we write The set [v, u] is a closed segment with end points u and v.The set [v, u⌈ is a closed ray in X with end point v with direction u − v, which is a closed convex cone with vertex at v and is a special class of cones in X.The set ⌉u, v⌈ is a line in X passing through points v and u.
We conclude this section by recalling the following result (see [38]): Theorem 2.4.Let X be a uniformly convex and uniformly smooth Banach space and let C a nonempty, closed, and convex subset of X.For any y ∈ C, let x ∈ X\C be such that y = P C x.We define the inverse image of y under the metric projection P C : X → C by P −1 C (y) = {u ∈ X : P C (u) = y}.Then P −1 C (y) is a closed cone with vertex at y in X.However, P −1 C (y) is not convex, in general.

Dual Cones for the Metric Projection
A cone K in a vector space is said to pointed if K has vertex at θ , K ∩ (−K) = {θ }, and K = θ .Let H be a Hilbert space, and let K be a cone in H with vertex at v. We define the dual cone of K in H with respect to the metric projection P K by The dual cone has the following properties in Hilbert spaces (see Zarantonello [1]): (1) K ⊥ is a closed and convex cone in H with vertex at v.
(3) If K is a closed and convex cone, then K ⊥ and K are dual cones of each other.(4) If K is a closed, convex and pointed cone, then P K is positive homogeneous and x, P K x = P K x 2 , for all x ∈ H.
In this following, we extend the concept of a dual cone from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and derive their valuable properties.We will show that the properties (3) and (4) given above do not hold, in general, in Banach spaces.Definition 3.1.Let X be a Banach space, let the dual X * of X be strictly convex, and let K be a cone in X with vertex v.We define the dual cone with respect to the metric projection P by The following result shows that K ⊥ P is a cone in X, and K ⊥ P and K have the same vertex v.
Theorem 3.2.Let X be a Banach space, let the dual X * of X be strictly convex, and let K be a cone in X with vertex v.Then, the following statements hold: (a) K ⊥ P is a cone with vertex at θ in X.(b) If X is uniformly convex and uniformly smooth, then K ⊥ P is closed.(c) If X is uniformly convex and uniformly smooth and K is closed and convex, then K Proof.(a) For an arbitrary x ∈ K ⊥ P with x = v, and for any t > 0, by the homogeneity property of the normalized duality mapping J, we have P , for all t > 0. Thus, K ⊥ P is a cone in X with vertex at v. (b) Under the additional hypothesis on X, J is continuous, which proves that K ⊥ P is closed.(c) By the basic variational principle of P K , for any given x ∈ X, we have Since ( 7) coincides with ( 6), we deduce that By using (2.1), we have Next, we compute , and hence (1, −1, 0).
Proposition 3.3.Let X be a uniformly convex and uniformly smooth Banach space and let K be a closed, convex, and pointed cone in X.Then P K is positive homogeneous.In general, Proof.For any t > 0, since K is a closed, convex and pointed cone in X, for any x ∈ X, we have and by appealing to the basic variational property of P K , this implies that P K (tx) = tP K x.

Generalized dual cones with respect to the generalized projection π
We now study the generalized dual cone of K for the generalized projection π.We first recall some properties of the inverse image of the vertex of a cone by the generalized projection π C in X * .
Given a uniformly convex and uniformly smooth Banach space X with dual space X * and a cone K with vertex at v, we recall that Theorem 4.1.Let X be a uniformly convex and uniformly smooth Banach space with dual X * and let K be a closed and convex cone in X with vertex at v.Then, Proof.(a) See Theorem 2.4.(b) For a fixed z ∈ K, we have By the basic variational principle for π π −1 K (v) , we obtain that z ∈ π −1 For the converse, for any z ∈ π −1 We recall that for ℓ ∈ X * , we have For the given z ∈ π −1 Then, for any x ∈ K, we define Since K is a closed and convex cone with vertex v, for all x ∈ K, we have that w ∈ K.By the variational principle for P K , we have which implies that Then ψ ∈ X * .By (12), we have By ( 11) and ( 14), we have (Jv), by the variational principle, we have Then, using (13), we have (Jv), we obtain π −1 (Jv) ⊆ K. This, in view of (10), completes the proof.
Definition 4.2.Let X be a uniformly convex and uniformly smooth Banach space with dual X * , and let K be a cone in X with vertex at v. We define the generalized dual cone of K in X * with respect to the generalized projection π by Theorem 4.3.Let X be a uniformly convex and uniformly smooth Banach space with dual X * , and let K be a cone in X with vertex at v. Then the following statements hold: π is a closed and convex cone with vertex at Jv in X * .(c) K and K ⊥ π are generalized dual of each other: Proof.(a).By the basic variational principle for π K , for any ψ ∈ X * and v ∈ K, we have Corollary 4.4.Let X be a uniformly convex and uniformly smooth Banach space, and let C and K be closed and convex cones in X with a common vertex at v satisfying C ∩ K = {v}.Then, π is a closed and convex cone with vertex at Jv, it follows that . By (c) of Theorem 4.3 and (a), we have On the other hand, from . Thus, by ( 16), we have which proves the desired identity.Since (C ∩ K) ⊥ π and co(C ⊥ π ∪ K ⊥ π ) are both closed and convex cones with vertex at Jv, we have the result by using Theorem 4.3.
The following result can be proved in an analogous fashion: Corollary 4.5.Let X be a uniformly convex and uniformly smooth Banach space, and let {K λ : λ ∈ Λ } be a set of closed and convex cones with a common vertex at v such that ∩ λ ∈Λ K λ = {v}.Then , where Λ is an arbitrary given index set.
Example 5.4.Let (S, A , µ) be a measure space with µ(S) ≥ 1.For any p ∈ [1, ∞), let X = L p (S) be the real Banach space of real functions defined on S with norm • p .For any given M > 0, let Then C is a nonempty, closed, and convex subset in L p (S).For any A ∈ A with 1 ≤ µ(A) < ∞, let 1 A denote the characteristic function of A, which satisfies 1 A ∈ L q (S) * = L p (S), where p, q ∈ [1, ∞] are such that 1 p + 1 q = 1.Then F C (1 A ) is a nonempty, closed, and convex subset of C such that For any f ∈ C, we have Thus, ( 18) and ( 19) imply that For the converse, we define h on S by By 1 ≤ µ(A) < ∞, we have By the above equation, it follows that for any g ∈ F C (1 A ) ⊆ C, we must have It follows that 1 A , g = M, that is, A g(s)dµ(s) = M.This implies that By combining (20) and ( 22), we get (17).By ( 21) and ( 20), we have h ∈ F C (1 A ), which shows that F C (1 A ) = / 0. This prove the claim.
Example 5.5.For any p with 1 ≤ p < ∞, let X = ℓ p be the real Banach space of real sequences with norm • p .For any given M > 0, let C = {x ∈ ℓ P : x p ≤ M}.
Then C is a nonempty, closed and convex subset in ℓ p .For any positive integers m and n with n ≥ 1.We define Then, F C (ℓ n m ) is a nonempty, closed, and convex subset of C such that Proof.We only need to show that F C (ℓ n m ) is nonempty.For this, we take z = {z i } ∈ ℓ p as follows: Then, it is easy to verify that z ∈ C and z ∈ F C (ℓ n m ).The rest of the arguments are similar to the ones used in Example 5.4.Lemma 5.6.Let X be a reflexive Banach space with dual space X * and let C be a closed, convex and bounded set in X.Then for each ψ ∈ X * , F C (ψ) is nonempty, closed, and convex subset of C.
Proof.Since C is weakly compact, for any ψ ∈ X * , the function ψ, • attains its maximum value on C. That is, there is y ∈ C such that ψ, y = max x∈C ψ, x .This implies that y ∈ F C (ψ).The set F C (ψ) is clearly, closed and convex.
Theorem 5.7.Let X be a reflexive Banach space with dual X * and let C be a nonempty, closed, and convex set in X.Then (a) For any u ∈ X, Proof.(a) For an arbitrary z ∈ C, by the basic variational principle for P C , we have This proves the first equality in (a).To prove the second inequality, for any z ∈ C, by the basic variational principle of π C , we have which proves the second equality in (a).(b) For any ψ ∈ X * , J * ψ ∈ X by substituting J * ψ for u ∈ X in (a) and noticing JJ * ψ = ψ, (b) follows at once.The conclusion of Theorem 5.7 can be described by the form of variational inequalities.
Corollary 5.8.Let X be a uniformly convex and uniformly smooth Banach space wth dual X * and let C be a nonempty, closed, and convex set in X.Then (a) For any u ∈ X, a point y ∈ C is a solution of the variational inequality Ju, y − x ≥ 0, for all x ∈ C, if and only if, y is a solution to one of the following projection equations: (b) For any ψ ∈ X * , a point y ∈ C is a solution of the variational inequality ψ, y − x ≥ 0, for all x ∈ C, if and only if, y is a solution to one of the following projection equations: y = P C (J * ψ + y) or y = π C (ψ + jy).

Visions in Banach spaces
Definition 5.9.Let X be a Banach space with dual X * and let C ⊂ X be nonempty, closed, and convex.
(a) We define the vision F −1 C (y) in X * of a point y ∈ C with respect to the background C by (b) We define the vision F −2 C (y) in X of a point y ∈ C with respect to the background C by Lemma 5.10.Let X be a uniformly convex and uniformly smooth Banach space with dual X * , and let C be a nonempty, closed, and convex subset in X.Then, for any y ∈ C, we have Since in a uniformly convex and uniformly smooth Banach space X, J and J * are both one-to-one and onto mapping such that J * J = I X and JJ * = I X * , the conclusions are evident.Proposition 5.11.Let X be a Banach space with dual X * and let C ⊂ X be nonempty, closed, and convex.Then, for any y ∈ C, we have which proves that ψ ∈ F −1 C (y), and hence F −1 C (y) is closed in X * .Proposition 5.12.Let X be a Banach space with dual space X * and let C be a nonempty, closed, and convex subset in X.Then for any y ∈ C, we have An analogue of the above result can be given by using the metric projection P C .Corollary 5.16.Let X be a uniformly convex and uniformly smooth Banach space with dual X * and let C be a nonempty, closed, and convex set in X.For any y ∈ C, we have Next we give some examples to demonstrate the concepts of J (C) and C (C).
Corollary 5.17.Let X be Banach space with dual X * and let C be a proper closed subspace of X. Then: Proof.Since C is a proper closed subspace of X, by the Hahn-Banach space theorem, there is ψ ∈ X * with ψ X * = 1 such that ψ, x = 0 for all x ∈ C.This implies that ψ ∈ F −1 C (y) for all y ∈ C. Since ψ = θ * , it follows at once that y ∈ C (C), for all y ∈ C. The claim then follows from Corollary 5.14.
In the following result, we use the closed and open balls and the unit sphere, see Section 2.
So, we may assume that ψ X * = r.It follows that J * ψ X = r, which implies J * ψ ∈ S(r).By y ∈ S(r), ψ ∈ F −1 B(r) (y) and J * ψ ∈ S(r), it follows that This implies ψ X * = y X = r and ψ, y = r 2 Hence ψ ∈ Jy.We have established, By combining (30) and (32), we complete the proof of (c).(d) It follows at once from (c) under the additional hypothesis on X.
The following result connects generalized dual cones with the notion of visions.
Theorem 5.19.Let X be a uniformly convex and uniformly smooth Banach space and let K be a closed and convex cone in X with vertex at v.Then, K Proof.By Theorem 4.3, we have K ⊥ π = π −1 K (v).Thus, we only need to prove the second equality in (33).For any ψ ∈ X * , we have and the proof is complete.
Remark 5.20.Equation (33) reexamines the following results: (i) K ⊥ π is a closed and convex cone with vertex at Jv in X * (Theorem 4.3).(ii) F −1 K (v) is a closed and convex cone with vertex at θ * in X * (Proposition 5.12).
which, due to the definition of K ⊥ π and (9) at once implies (a).(b) Since π −1 K (v) is a closed and convex cone with vertex at Jv in X * , (b) follows at once (a).Finally, (c) follows from (a) and Theorem 4.1.
evident.The converse follows from part (c) of Theorem 4.3.(b) It follows at once that C ∩ K is a closed and convex cone in X with vertex v.By (a), the inclusion C