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Maximal monotone normal cones in locally convex spaces
Accepted Manuscript Maximal monotone normal cones in locally convex spaces
M.D. Voisei
PII: DOI: Reference:
S0022-247X(19)30324-5 https://doi.org/10.1016/j.jmaa.2019.04.017 YJMAA 23105
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
4 March 2019
Please cite this article in press as: M.D. Voisei, Maximal monotone normal cones in locally convex spaces, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.04.017
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Maximal monotone normal cones in locally convex spaces M.D. Voisei Abstract Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop-Phelps and sum representability results are presented in the last part.
1
Preliminaries
The aim of this paper is to characterize the subsets C of a locally convex space (X, τ ) whose normal cone NC is a maximal monotone operator. Here (X, τ ) is a non-trivial (that is, X = {0}) real Hausdorff separated locally convex space (LCS for short), X ∗ is its topological dual usually endowed with the weak-star topology denoted by w∗ , (X ∗ , w∗ )∗ is identified with X, x, x∗ := x∗ (x) =: c(x, x∗ ), for x ∈ X, x∗ ∈ X ∗ denotes the duality product or coupling of X × X ∗ , and Graph T = {(x, x∗ ) ∈ X × X ∗ | x∗ ∈ T (x)} stands for the graph of T : X ⇒ X ∗ . Rockafellar showed in [3, Theorem A] that when X is a Banach space and f : X → R is proper convex lower semicontinuous then its convex subdifferential ∂f : X ⇒ X ∗ , defined by x∗ ∈ ∂f (x) if f (x) is finite and for every y ∈ X, f (y) ≥ f (x) + y − x, x∗ , is maximal monotone (∂f ∈ M(X) for short). In particular the normal cone to C which is given by NC = ∂ιC ∈ M(X), whenever C ⊂ X is closed convex. Here ιC (x) = 0, for x ∈ C; ιC (x) = +∞, for x ∈ X \ C denotes the indicator function of C. The motivation for studying the maximal monotonicity of a normal cone in the general context of locally convex spaces comes from the connection of this problem with the separation theorem (see Theorem 3 below) and its applications to Bishop-Phelps type results (see Theorems 7, 9 and Corollary 12 below) and, more importantly, to the representability of the sum of a maximal monotone operator and a normal cone (see Theorem 16 below) because representability is the first step towards maximal monotonicity and because of the universality the normal cone holds in the sum theorem for maximal monotone operators. Therefore it is very interesting to find when a normal cone is maximal monotone outside the Banach space context. Our main argument stems from the explicit form of the normal cone Fitzpatrick function (see Theorem 1 below) and our characterization of maximal monotone operators as representable and of type NI (see [4, Theorem 2.3] or [5, Theorem 3.4]). 1
Recall that the Fitzpatrick function ϕT : X × X ∗ → R of a multi-valued operator T : X ⇒ X ∗ is given by (see [2]) ϕT (x, x∗ ) := sup{x − a, a∗ + a, x∗ | (a, a∗ ) ∈ Graph T }, (x, x∗ ) ∈ X × X ∗ .
(1)
As usual, given a LCS (E, μ) and A ⊂ E we denote by “conv A” the convex hull of A, μ “span A” the linear hull of A, “clμ (A) = A ” the μ−closure of A, “intμ A” the μ−topological interior of A, “core A” the algebraic interior of A. The use of the μ−notation is not enforced when the topology μ is clearly understood. For f, g : E → R we set [f ≤ g] := {x ∈ E | f (x) ≤ g(x)}; the sets [f = g], [f < g], and [f > g] being defined in a similar manner. We write f ≥ g shorter for f (z) ≥ g(z), for every z ∈ E. For a multi-function T : X ⇒ X ∗ , D(T ) = PrX (Graph T ), R(T ) = PrX ∗ (Graph T ) stand for the domain and the range of T respectively, where PrX , PrX ∗ denote the projections of X × X ∗ onto X, X ∗ respectively. When no confusion can occur, T : X ⇒ X ∗ will be identified with Graph T ⊂ X × X ∗ . The restriction of an operator T : X ⇒ X ∗ to U ⊂ X is the operator T |U : X ⇒ X ∗ defined by Graph(T |U ) = Graph T ∩ (U × X ∗ ). The operator T + : X ⇒ X ∗ whose graph is Graph(T + ) := [ϕT ≤ c] describes all (x, x∗ ) ∈ X × X ∗ that are monotonically related (m.r. for short) to T , that is (x, x∗ ) ∈ [ϕT ≤ c] iff, for every (a, a∗ ) ∈ Graph T , x − a, x∗ − a∗ ≥ 0. We consider the following classes of functions and operators on (X, τ ) Λ(X) the class formed by proper convex functions f : X → R. Recall that f is proper if dom f := {x ∈ X | f (x) < ∞} is nonempty and f does not take the value −∞, Γτ (X) the class of functions f ∈ Λ(X) that are τ –lower semi-continuous (τ –lsc for short), M(X) the class of non-empty monotone operators T : X ⇒ X ∗ (Graph T = ∅). Recall that T : X ⇒ X ∗ is monotone if, for all (x1 , x∗1 ), (x2 , x∗2 ) ∈ Graph T , x1 − x2 , x∗1 − x∗2 ≥ 0 or, equivalently, Graph T ⊂ Graph(T + ). M(X) the class of maximal monotone operators T : X ⇒ X ∗ . The maximality is understood in the sense of graph inclusion as subsets of X × X ∗ . It is easily seen that T ∈ M(X) iff T = T + . To a proper function f : (X, τ ) → R we associate the following notions: Epi f := {(x, t) ∈ X × R | f (x) ≤ t} is the epigraph of f , conv f : X → R, the convex hull of f , which is the greatest convex function majorized by f , (conv f )(x) := inf{t ∈ R | (x, t) ∈ conv(Epi f )} for x ∈ X, clτ conv f : X → R, the τ −lsc convex hull of f , which is the greatest τ –lsc convex function majorized by f , (clτ conv f )(x) := inf{t ∈ R | (x, t) ∈ clτ (conv Epi f )} for x ∈ X, 2
f ∗ : X ∗ → R is the convex conjugate of f : X → R with respect to the dual system (X, X ∗ ), f ∗ (x∗ ) := sup{x, x∗ − f (x) | x ∈ X} for x∗ ∈ X ∗ . Accordingly, σC (x∗ ) := sup{x, x∗ | x ∈ C} = ι∗C (x∗ ), for x∗ ∈ X ∗ . Recall that f ∗∗ := (f ∗ )∗ = cl conv f whenever cl conv f (or equivalently f ∗ ) is proper, where for functions defined in X ∗ , the conjugates are taken with respect to the dual system (X ∗ , X). Throughout this article the conventions ∞ − ∞ = ∞, sup ∅ = −∞, and inf ∅ = ∞ are enforced while the use of the topology notation is avoided when the topology is clearly understood. All the considerations and results of this paper can be done with respect to a separated dual system of vector spaces (X, Y ).
2
Support points and the maximality of the normal cone
Given (X, τ ) a LCS and C ⊂ X, we denote by Supp C = {x ∈ C | NC (x) = {0}} the set of support points of C and by C # := {x ∈ X | ∀(a, a∗ ) ∈ Graph NC , x − a, a∗ ≤ 0} the portable hull of C which is the intersection of all the supporting half-spaces that contain C and are supported at points in C.
C
C#
Supp C
From their definitions, Supp ∅ = Supp X = ∅, ∅# = X, and x ∈ C # ⇔ ∀a ∈ Supp C, ∀a∗ ∈ NC (a), x − a, a∗ ≤ 0, with the remarks that, for every C ⊂ X, one has C # = ∅ and clτ conv C ⊂ (clτ conv C)# ⊂ C # ; while C # = X when Supp C = ∅, e.g., X # = X.
3
Theorem 1 Let X be a LCS. For every C ⊂ X, (x, x∗ ) ∈ X × X ∗ ϕNC (x, x∗ ) = ιC # (x) + σC (x∗ ).
(2)
In particular, for every ∅ = C ⊂ X, (x, x∗ ) ∈ X × X ∗ , we have ϕNC (x, 0) = ιC # (x) ≥ 0, ϕNC (x, x∗ ) ≤ ιC (x) + σC (x∗ ) and C # := PrX (dom ϕNC ) = {x ∈ X | ϕNC (x, 0) = (≤)0} = D(NC+ ).
(3)
Proof. For C = ∅ relation (2) is straightforward. Otherwise, because for every a ∈ C, NC (a) is a cone we get that, for every (x, x∗ ) ∈ X × X ∗ , ϕNC (x, x∗ )
= sup{x − a, n∗ + a, x∗ | a ∈ C, n∗ ∈ NC (a)} ≤ sup{x − a, n∗ | a ∈ C, n∗ ∈ NC (a)} + sup{a, x∗ | a ∈ C} = ιC # (x) + σC (x∗ ).
To conclude, it suffices to show that, for every (x, x∗ ) ∈ dom ϕNC , ϕNC (x, x∗ ) ≥ ιC # (x)+ σC (x∗ ). But for (x, x∗ ) ∈ dom ϕNC we have x ∈ C # and ϕNC (x, x∗ ) ≥ σC (x∗ ). The last part of Theorem 1 says that any monotone extension T of NC has D(T ) ⊂ C # or that we can extend NC monotonically only inside C # × X ∗ . Theorem 2 Let X be a LCS and let C ⊂ X. The following are equivalent (i) NC ∈ M(X), (ii) ϕNC (x, x∗ ) = ιC (x) + σC (x∗ ), (x, x∗ ) ∈ X × X ∗ , (iii) C # ⊂ (=)C (iv) C = {x ∈ X | ϕNC (x, 0) ≤ 0}. Proof. (i) ⇒ (iii) Since NC ∈ M(X), we know that C = ∅, NC = NC+ and so C # = D(NC+ ) = D(NC ) = C. (iii) ⇔ (iv) follows from the fact that both (iii) and (iv) imply that C is non-empty in which case ϕNC (x, 0) = ιC # (x), x ∈ X. (iii) ⇒ (ii) follows directly from (2). (ii) ⇒ (i) According to our conventions, the equality ϕNC (x, x∗ ) = ιC (x) + σC (x∗ ), (x, x∗ ) ∈ X × X ∗ , implies that C is non-empty convex and NC+ = [ϕNC ≤ c] ⊂ NC , that is NC ∈ M(X). Concerning the previous result note that C is non-empty closed convex whenever NC ∈ M(X). Also, subpoint (iv) can be restated equivalently as (iv) C is non-empty and C = (⊃){x ∈ X | ϕNC (x, 0) ≤ (=)0}. Theorem 2 has strong ties with the separation theorem. Assume that C ⊂ X is closed and convex. Then for every x ∈ X \ C there is n∗ ∈ X ∗ \ {0} such that x, n∗ > σC (n∗ ) = 4
sup{u, n∗ | u ∈ C}. In the next theorem we see that the maximality of NC is equivalent to the possibility of picking, in the previous separation argument, of a non-zero n∗ that attains its global maximum on C, that is, n∗ ∈ R(NC ) which is also called a support functional of C (see e.g. [1]). Theorem 3 Let X be a LCS and let C ⊂ X. Then NC ∈ M(X) (or C = C # ) iff for every x ∈ X \ C there is n∗ ∈ D(∂σC ) = R(NC ) such that x, n∗ > σC (n∗ ). Proof. We have ∀x ∈ X \ C, ∃(y, n∗ ) ∈ NC : x − y, n∗ > 0 ⇔ (x ∈ X \ C ⇒ x ∈ X \ C # ) ⇔ C # ⊂ C ⇔ NC ∈ M(X).
Corollary 4 Let X be a LCS and let C ⊂ X be non-empty closed and convex. If X is a Banach space or int C = ∅, or C is weakly compact then NC ∈ M(X). Proof. When X is a Banach space or int C = ∅ it is known that C # = C and we may apply Theorem 2. If C is weakly compact then every x∗ ∈ X ∗ attains its global maximum on C (i.e., R(NC ) = X ∗ ) and we use Theorem 3. Note that C # , the portable hull of C, is the smallest set formed by intersecting halfspaces that contain C and are supported at points in C, and, at the same time, the largest set on which the normal cone NC can be extended monotonically. Proposition 5 Let X be a LCS. For every C ⊂ X, NC # |C = NC , Graph NC ⊂ Graph NC # , NC # ∈ M(X), and C ## := (C # )# = C # . Proof. The inclusion Graph NC ⊂ Graph NC # follows directly from the definition of C # while Graph(NC # |C ) ⊂ Graph NC is plain. To conclude it suffices to prove that C ## ⊂ C # . To this end, note that, for every x ∈ C ## and for every (a, a∗ ) ∈ NC # , we have x − a, a∗ ≤ 0. In particular, for (a, a∗ ) ∈ NC , we get x ∈ C # . Remark 6 Every (maximal) monotone extension of NC has the domain contained in C # , that is, NC ⊂ T ∈ M(X) ⇒ D(T ) ⊂ D(NC+ ) = C # . Therefore NC # is a maximal monotone extension of NC with the largest possible domain. In general, NC # is not the only maximal monotone extension of NC (even with the largest possible domain or with a normal cone structure). For example, for C = (0, 1] ⊂ R, NC admits NC and NC # as two different maximal monotone extensions; moreover NC has an infinity of maximal monotone extensions with the largest possible domain C # = (−∞, 1]. The maximality of the subdifferential allows us to reprove and extend some of the Bishop-Phelps results (see [1]). 5
Theorem 7 Let X be a LCS. If, for every closed convex C ⊂ X, NC ∈ M(X), then, for every closed convex C ⊂ X, Supp C is dense in bd C. Proof. If int C = ∅ then Supp C = bd C and we are done. In case int C = ∅ we have C = bd C and we prove that, for every closed convex U such that C ∩ int U = ∅, Supp C ∩ U = ∅. Let U be closed convex such that C ∩ int U = ∅. Assume, by contradiction, that NC |U (x) = {0}, for every x ∈ C ∩ U . Then M(X) NC∩U = NC + NU = NU |C ⊂ NU ∈ M(X), which provides the contradiction U ⊂ C, i.e., int C = ∅. Remark 8 The previous results still holds for a fixed closed convex C ⊂ X if, for every U ⊂ X closed convex such that C ∩ int U = ∅, NC∩U ∈ M(X). Theorem 9 Let X be a LCS and let C ⊂ X be closed convex, free of lines, and finitedimensional, that is, dim(span C) < ∞. Then NC ∈ M(X) and clw∗ R(NC ) = clw∗ (dom σC ). If, in addition, C is bounded, then clw∗ R(NC ) = X ∗ . Proof. For every x ∈ C the set [σC−x < 0] is non-empty. Under the given assumptions σC is w∗ −continuous on intw∗ (dom σC ) and intw∗ [σC−x < 0] = ∅. For every x∗ ∈ clw∗ (dom σC ) and every weak-star closed convex neighborhood V of x∗ , σC + ιV ∈ Γw∗ (X ∗ ), and σC + ιV is w∗ −continuous on intw∗ (dom σC ∩ V ); whence ∂(σC + ιV ) = ∂σC + NV and Graph(∂(σC + ιV )) = ∅. That yields V ∩ D(∂σC ) = ∅ and, as a consequence, clw∗ R(NC ) = clw∗ (dom σC ). From [σC−x < 0] ⊂ dom σC ⊂ cl w∗ R(NC ) and intw∗ [σC−x < 0] = ∅ we know that [σC−x < 0] ∩ R(NC ) = ∅. According to Theorem 3, NC ∈ M(X). If, in addition, C is bounded, then dom σC = X ∗ . Proposition 10 Let X be a LCS. If, for every f ∈ Γ(X), Graph ∂f = ∅, then for every f ∈ Γ(X), D(∂f ) is dense in dom f and R(∂f ) is weak-star dense in dom f ∗ . Proof. For every x ∈ dom f and every closed convex neighborhood V of x, f + ιV ∈ Γ(X); whence Graph ∂(f + ιV ) = ∅. Since ιV is continuous at x ∈ dom f , ∂(f + ιV ) = ∂f + ∂ιV ; in particular V ∩ D(∂f ) = ∅. Corollary 11 Let X be a LCS such that for every f ∈ Γ(X), Graph ∂f = ∅. Then for every closed convex bounded C ⊂ X the support functionals of C are weak-star dense in X ∗. Corollary 12 Let (X, · ) be a Banach space. Then for every closed convex bounded C ⊂ X the support functionals of C are strongly dense in X ∗ . If C ⊂ X is merely closed and convex then the support functionals of C are strongly dense in the domain of σC . In other words for every > 0, f ∈ X ∗ such that supC f < +∞ there is g ∈ X ∗ , x0 ∈ C such that g(x0 ) = supC g (that is, g ∈ NC (x0 )) and f − g ≤ . 6
Proof. Since X ∗ is a Banach space, for every f ∈ Γ(X ∗ ), ∂f ∈ M(X ∗ ); in particular Graph ∂f = ∅. According to our previous proposition D(∂σC ) is strongly dense in dom σC . Here “∂” denotes the subdifferential of functions defined in X ∗ with respect to the duality (X ∗ , X ∗∗ ). Note that the set of support functionals represent D(∂w∗ σC ) where “∂w∗ ” denotes the subdifferential of functions defined in X ∗ with respect to the duality (X ∗ , X). It suffices to show that D(∂w∗ σC ) is strongly dense in D(∂σC ) fact already observed in [3, Proposition 1, p. 211]. Again, if in addition, C is closed then dom σC = X ∗ .
3
Partial portable hulls and sum representability
Let (X, τ ) be a LCS and C, S ⊂ X. We denote by CS# the partial portable hull of C on S which is the intersection of all the (supporting) half-spaces that contain C and are supported at points in S, CS# := {x ∈ X | ∀a ∈ C ∩ S, a∗ ∈ NC (a), x − a, a∗ ≤ 0}. Equivalently x ∈ CS# ⇔ ∀a ∈ S ∩ Supp C, a∗ ∈ NC (a), x − a, a∗ ≤ 0, with the remarks that CS# = X when S ∩ Supp C = ∅ and that, for every C, S ⊂ X, # # CS# = ∅, C ⊂ clτ conv C ⊂ C # ⊂ CS# , CS# = CS∩C = CS∩Supp C. # # Note also that CS = C whenever S ⊃ Supp C.
C
CS#
S ∩ Supp C
A more comprehensive definition of the portable hull of a set C ⊂ X can be given with respect to G ⊂ Graph(NC ) by CG# := {x ∈ X | ∀(a, a∗ ) ∈ G, x − a, a∗ ≤ 0}, # with the remark that CS# = CGraph(N . C |S )
7
(4)
Proposition 13 Let X be a LCS and let C, S ⊂ X. Then (i) Graph(NC |S ) ⊂ Graph NC # and S
NC |S = NC # |S iff S ∩ C = S ∩ CS# . S
In particular NC # |C = NC . # # # (ii) (CS# )# S = (CS ) = CS . In particular NC # ∈ M(X). S
Proof. (i) For every a ∈ S ∩ C, a∗ ∈ NC (a), x ∈ CS# , x − a, a∗ ≤ 0, that is a∗ ∈ NC # (a); S whence NC |S ⊂ NC # |S . For the second part it suffices to prove the converse implication. S
Let x ∈ S ∩ CS# = S ∩ C, x∗ ∈ NC # (x), that is, for every y ∈ CS# , y − x, x∗ ≤ 0. For S y ∈ C we find x∗ ∈ NC (x). In particular, for S = C we have C ∩ C # = C so NC # |C = NC . ∗ (ii) For every x ∈ (CS# )# (S) and for every (a, a ) ∈ Graph(NC # ), (a ∈ S), we have S
x − a, a∗ ≤ 0. In particular, for (a, a∗ ) ∈ Graph(NC |S ), we get x ∈ CS# . We proved that # (CS# )# (S) ⊂ CS . The converse inclusion is immediate. Theorem 14 Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X. Then ϕT +NC = ϕT |C + ιC #
D(T )
×X ∗ .
(5)
Proof. According to its definition ϕT +NC (x, x∗ ) = sup{x − a, a∗ + a, x∗ + x − a, n∗ | a ∈ D(T ) ∩ C, a∗ ∈ T (a), n∗ ∈ NC (a)}, (x, x∗ ) ∈ X × X ∗ . Since the values of the normal cone are cones this yields that, for every x ∈ PrX (dom ϕT +NC ), a ∈ D(T ) ∩ C, n∗ ∈ NC (a), # we have x − a, n∗ ≤ 0, that is, PrX (dom ϕT +NC ) ⊂ CD(T ) ; from which one obtains, after ∗ taking n = 0, that ∀(x, x∗ ) ∈ X × X ∗ , ϕT +NC (x, x∗ ) ≥ ϕT |C (x, x∗ ) + ιC # (x). D(T )
The converse inequality is plain. Recall the following notion (see Definition 14 in [6]) Definition 15 Let (X, τ ) be a LCS. An operator T : X ⇒ X ∗ is representable in C ⊂ X or C−representable if C ∩ D(T ) = ∅ and there is h ∈ Γτ ×w∗ (X × X ∗ ) such that h ≥ c and [h = c] ∩ C × X ∗ = Graph(T |C ). Using the partial portable hull we can recover the representability of the sum between a representable operator and the normal cone (see Theorem 35 in [6]) Theorem 16 Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X be closed convex such that D(T ) ∩ int C = ∅. Then ψT +NC = ψT |C 2 σC # ×{0} or D(T )
∗
∗
∗
ψT +NC (x, x ) = min{ψT |C (x, x − u ) + σC # (u∗ ) | u∗ ∈ X ∗ }, (x, x∗ ) ∈ X × X ∗ , D(T )
[ψT +NC = c] = [ψT |C = c] + NC . In particular, if, in addition, T is C−representable then T + NC is representable. 8
(6) (7)
Proof. The identity in (6) follows from ϕT +NC = ϕT |C + ιC #
D(T )
ιC #
D(T )
×X ∗
×X ∗
and the fact that
# ∗ is continuous on int CD(T ) × X (see e.g. [7, Theorem 2.8.7(iii)]).
For (7) note first that dom ψT +NC ⊂ C × X ∗ . Hence (x, x∗ ) ∈ [ψT +NC = c] iff there is u ∈ X ∗ such that ψT |C (x, x∗ − u∗ ) + σC # (u∗ ) = x, x∗ iff (x, x∗ − u∗ ) ∈ [ψT |C = c] and ∗
(x, u∗ ) ∈ NC # .
D(T )
D(T )
If, in addition, T is (C−)representable then, using Theorem 16 in [6] and (7), we get [ψT +NC = c] = T + NC , that is, T + NC is representable.
4
Concluding remarks
Let (X, τ ) be a LCS and let us call a set C ⊂ X portable if C = C # (or NC ∈ M(X)). Some of the results of this paper can be summarized as follows: • C ⊂ X is portable iff for every x ∈ X \ C there is n∗ ∈ D(∂σC ) = R(NC ) such that x, n∗ > σC (n∗ ) or, equivalently, [σC−x < 0] ∩ R(NC ) = ∅. • If C ⊂ X is closed convex and X is a Banach space or int C = ∅, or C is weakly compact then C is portable. • For every C, S ⊂ X, CS# is portable. • If every closed convex C ⊂ X is portable then for every closed convex C ⊂ X, Supp C is dense in bd C. • Every closed convex, free of lines, and finite-dimensional set is portable. The results in this article are an incentive for studying the following problems: (P1) Find characterizations of all LCS’s X with the property that, for every closed convex C ⊂ X, NC ∈ M(X). (P2) Find characterizations of all LCS’s X with the property that, for every f ∈ Γ(X), Graph ∂f = ∅. (P3) Find characterizations of all LCS’s X with the property that, for every f ∈ Γ(X), ∂f ∈ M(X).
References [1] Errett Bishop and R. R. Phelps. The support functionals of a convex set. In Proc. Sympos. Pure Math., Vol. VII, pages 27–35. Amer. Math. Soc., Providence, R.I., 1963.
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[2] Simon Fitzpatrick. Representing monotone operators by convex functions. In Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), volume 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 59–65. Austral. Nat. Univ., Canberra, 1988. [3] R. T. Rockafellar. On the maximal monotonicity of subdifferential mappings. Pacific J. Math., 33:209–216, 1970. [4] M. D. Voisei. A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces. Math. Sci. Res. J., 10(2):36–41, 2006. [5] M. D. Voisei. The sum and chain rules for maximal monotone operators. Set-Valued Anal., 16(4):461–476, 2008. [6] M. D. Voisei. Location, Identification, and Representability of Monotone Operators in Locally Convex Spaces. Set-Valued Var. Anal., 27(1):151–168, 2019. [7] C. Zălinescu. Convex analysis in general vector spaces. World Scientific Publishing Co. Inc., River Edge, NJ, 2002.
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M.D. Voisei
PII: DOI: Reference:
S0022-247X(19)30324-5 https://doi.org/10.1016/j.jmaa.2019.04.017 YJMAA 23105
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
4 March 2019
Please cite this article in press as: M.D. Voisei, Maximal monotone normal cones in locally convex spaces, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.04.017
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Maximal monotone normal cones in locally convex spaces M.D. Voisei Abstract Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop-Phelps and sum representability results are presented in the last part.
1
Preliminaries
The aim of this paper is to characterize the subsets C of a locally convex space (X, τ ) whose normal cone NC is a maximal monotone operator. Here (X, τ ) is a non-trivial (that is, X = {0}) real Hausdorff separated locally convex space (LCS for short), X ∗ is its topological dual usually endowed with the weak-star topology denoted by w∗ , (X ∗ , w∗ )∗ is identified with X, x, x∗ := x∗ (x) =: c(x, x∗ ), for x ∈ X, x∗ ∈ X ∗ denotes the duality product or coupling of X × X ∗ , and Graph T = {(x, x∗ ) ∈ X × X ∗ | x∗ ∈ T (x)} stands for the graph of T : X ⇒ X ∗ . Rockafellar showed in [3, Theorem A] that when X is a Banach space and f : X → R is proper convex lower semicontinuous then its convex subdifferential ∂f : X ⇒ X ∗ , defined by x∗ ∈ ∂f (x) if f (x) is finite and for every y ∈ X, f (y) ≥ f (x) + y − x, x∗ , is maximal monotone (∂f ∈ M(X) for short). In particular the normal cone to C which is given by NC = ∂ιC ∈ M(X), whenever C ⊂ X is closed convex. Here ιC (x) = 0, for x ∈ C; ιC (x) = +∞, for x ∈ X \ C denotes the indicator function of C. The motivation for studying the maximal monotonicity of a normal cone in the general context of locally convex spaces comes from the connection of this problem with the separation theorem (see Theorem 3 below) and its applications to Bishop-Phelps type results (see Theorems 7, 9 and Corollary 12 below) and, more importantly, to the representability of the sum of a maximal monotone operator and a normal cone (see Theorem 16 below) because representability is the first step towards maximal monotonicity and because of the universality the normal cone holds in the sum theorem for maximal monotone operators. Therefore it is very interesting to find when a normal cone is maximal monotone outside the Banach space context. Our main argument stems from the explicit form of the normal cone Fitzpatrick function (see Theorem 1 below) and our characterization of maximal monotone operators as representable and of type NI (see [4, Theorem 2.3] or [5, Theorem 3.4]). 1
Recall that the Fitzpatrick function ϕT : X × X ∗ → R of a multi-valued operator T : X ⇒ X ∗ is given by (see [2]) ϕT (x, x∗ ) := sup{x − a, a∗ + a, x∗ | (a, a∗ ) ∈ Graph T }, (x, x∗ ) ∈ X × X ∗ .
(1)
As usual, given a LCS (E, μ) and A ⊂ E we denote by “conv A” the convex hull of A, μ “span A” the linear hull of A, “clμ (A) = A ” the μ−closure of A, “intμ A” the μ−topological interior of A, “core A” the algebraic interior of A. The use of the μ−notation is not enforced when the topology μ is clearly understood. For f, g : E → R we set [f ≤ g] := {x ∈ E | f (x) ≤ g(x)}; the sets [f = g], [f < g], and [f > g] being defined in a similar manner. We write f ≥ g shorter for f (z) ≥ g(z), for every z ∈ E. For a multi-function T : X ⇒ X ∗ , D(T ) = PrX (Graph T ), R(T ) = PrX ∗ (Graph T ) stand for the domain and the range of T respectively, where PrX , PrX ∗ denote the projections of X × X ∗ onto X, X ∗ respectively. When no confusion can occur, T : X ⇒ X ∗ will be identified with Graph T ⊂ X × X ∗ . The restriction of an operator T : X ⇒ X ∗ to U ⊂ X is the operator T |U : X ⇒ X ∗ defined by Graph(T |U ) = Graph T ∩ (U × X ∗ ). The operator T + : X ⇒ X ∗ whose graph is Graph(T + ) := [ϕT ≤ c] describes all (x, x∗ ) ∈ X × X ∗ that are monotonically related (m.r. for short) to T , that is (x, x∗ ) ∈ [ϕT ≤ c] iff, for every (a, a∗ ) ∈ Graph T , x − a, x∗ − a∗ ≥ 0. We consider the following classes of functions and operators on (X, τ ) Λ(X) the class formed by proper convex functions f : X → R. Recall that f is proper if dom f := {x ∈ X | f (x) < ∞} is nonempty and f does not take the value −∞, Γτ (X) the class of functions f ∈ Λ(X) that are τ –lower semi-continuous (τ –lsc for short), M(X) the class of non-empty monotone operators T : X ⇒ X ∗ (Graph T = ∅). Recall that T : X ⇒ X ∗ is monotone if, for all (x1 , x∗1 ), (x2 , x∗2 ) ∈ Graph T , x1 − x2 , x∗1 − x∗2 ≥ 0 or, equivalently, Graph T ⊂ Graph(T + ). M(X) the class of maximal monotone operators T : X ⇒ X ∗ . The maximality is understood in the sense of graph inclusion as subsets of X × X ∗ . It is easily seen that T ∈ M(X) iff T = T + . To a proper function f : (X, τ ) → R we associate the following notions: Epi f := {(x, t) ∈ X × R | f (x) ≤ t} is the epigraph of f , conv f : X → R, the convex hull of f , which is the greatest convex function majorized by f , (conv f )(x) := inf{t ∈ R | (x, t) ∈ conv(Epi f )} for x ∈ X, clτ conv f : X → R, the τ −lsc convex hull of f , which is the greatest τ –lsc convex function majorized by f , (clτ conv f )(x) := inf{t ∈ R | (x, t) ∈ clτ (conv Epi f )} for x ∈ X, 2
f ∗ : X ∗ → R is the convex conjugate of f : X → R with respect to the dual system (X, X ∗ ), f ∗ (x∗ ) := sup{x, x∗ − f (x) | x ∈ X} for x∗ ∈ X ∗ . Accordingly, σC (x∗ ) := sup{x, x∗ | x ∈ C} = ι∗C (x∗ ), for x∗ ∈ X ∗ . Recall that f ∗∗ := (f ∗ )∗ = cl conv f whenever cl conv f (or equivalently f ∗ ) is proper, where for functions defined in X ∗ , the conjugates are taken with respect to the dual system (X ∗ , X). Throughout this article the conventions ∞ − ∞ = ∞, sup ∅ = −∞, and inf ∅ = ∞ are enforced while the use of the topology notation is avoided when the topology is clearly understood. All the considerations and results of this paper can be done with respect to a separated dual system of vector spaces (X, Y ).
2
Support points and the maximality of the normal cone
Given (X, τ ) a LCS and C ⊂ X, we denote by Supp C = {x ∈ C | NC (x) = {0}} the set of support points of C and by C # := {x ∈ X | ∀(a, a∗ ) ∈ Graph NC , x − a, a∗ ≤ 0} the portable hull of C which is the intersection of all the supporting half-spaces that contain C and are supported at points in C.
C
C#
Supp C
From their definitions, Supp ∅ = Supp X = ∅, ∅# = X, and x ∈ C # ⇔ ∀a ∈ Supp C, ∀a∗ ∈ NC (a), x − a, a∗ ≤ 0, with the remarks that, for every C ⊂ X, one has C # = ∅ and clτ conv C ⊂ (clτ conv C)# ⊂ C # ; while C # = X when Supp C = ∅, e.g., X # = X.
3
Theorem 1 Let X be a LCS. For every C ⊂ X, (x, x∗ ) ∈ X × X ∗ ϕNC (x, x∗ ) = ιC # (x) + σC (x∗ ).
(2)
In particular, for every ∅ = C ⊂ X, (x, x∗ ) ∈ X × X ∗ , we have ϕNC (x, 0) = ιC # (x) ≥ 0, ϕNC (x, x∗ ) ≤ ιC (x) + σC (x∗ ) and C # := PrX (dom ϕNC ) = {x ∈ X | ϕNC (x, 0) = (≤)0} = D(NC+ ).
(3)
Proof. For C = ∅ relation (2) is straightforward. Otherwise, because for every a ∈ C, NC (a) is a cone we get that, for every (x, x∗ ) ∈ X × X ∗ , ϕNC (x, x∗ )
= sup{x − a, n∗ + a, x∗ | a ∈ C, n∗ ∈ NC (a)} ≤ sup{x − a, n∗ | a ∈ C, n∗ ∈ NC (a)} + sup{a, x∗ | a ∈ C} = ιC # (x) + σC (x∗ ).
To conclude, it suffices to show that, for every (x, x∗ ) ∈ dom ϕNC , ϕNC (x, x∗ ) ≥ ιC # (x)+ σC (x∗ ). But for (x, x∗ ) ∈ dom ϕNC we have x ∈ C # and ϕNC (x, x∗ ) ≥ σC (x∗ ). The last part of Theorem 1 says that any monotone extension T of NC has D(T ) ⊂ C # or that we can extend NC monotonically only inside C # × X ∗ . Theorem 2 Let X be a LCS and let C ⊂ X. The following are equivalent (i) NC ∈ M(X), (ii) ϕNC (x, x∗ ) = ιC (x) + σC (x∗ ), (x, x∗ ) ∈ X × X ∗ , (iii) C # ⊂ (=)C (iv) C = {x ∈ X | ϕNC (x, 0) ≤ 0}. Proof. (i) ⇒ (iii) Since NC ∈ M(X), we know that C = ∅, NC = NC+ and so C # = D(NC+ ) = D(NC ) = C. (iii) ⇔ (iv) follows from the fact that both (iii) and (iv) imply that C is non-empty in which case ϕNC (x, 0) = ιC # (x), x ∈ X. (iii) ⇒ (ii) follows directly from (2). (ii) ⇒ (i) According to our conventions, the equality ϕNC (x, x∗ ) = ιC (x) + σC (x∗ ), (x, x∗ ) ∈ X × X ∗ , implies that C is non-empty convex and NC+ = [ϕNC ≤ c] ⊂ NC , that is NC ∈ M(X). Concerning the previous result note that C is non-empty closed convex whenever NC ∈ M(X). Also, subpoint (iv) can be restated equivalently as (iv) C is non-empty and C = (⊃){x ∈ X | ϕNC (x, 0) ≤ (=)0}. Theorem 2 has strong ties with the separation theorem. Assume that C ⊂ X is closed and convex. Then for every x ∈ X \ C there is n∗ ∈ X ∗ \ {0} such that x, n∗ > σC (n∗ ) = 4
sup{u, n∗ | u ∈ C}. In the next theorem we see that the maximality of NC is equivalent to the possibility of picking, in the previous separation argument, of a non-zero n∗ that attains its global maximum on C, that is, n∗ ∈ R(NC ) which is also called a support functional of C (see e.g. [1]). Theorem 3 Let X be a LCS and let C ⊂ X. Then NC ∈ M(X) (or C = C # ) iff for every x ∈ X \ C there is n∗ ∈ D(∂σC ) = R(NC ) such that x, n∗ > σC (n∗ ). Proof. We have ∀x ∈ X \ C, ∃(y, n∗ ) ∈ NC : x − y, n∗ > 0 ⇔ (x ∈ X \ C ⇒ x ∈ X \ C # ) ⇔ C # ⊂ C ⇔ NC ∈ M(X).
Corollary 4 Let X be a LCS and let C ⊂ X be non-empty closed and convex. If X is a Banach space or int C = ∅, or C is weakly compact then NC ∈ M(X). Proof. When X is a Banach space or int C = ∅ it is known that C # = C and we may apply Theorem 2. If C is weakly compact then every x∗ ∈ X ∗ attains its global maximum on C (i.e., R(NC ) = X ∗ ) and we use Theorem 3. Note that C # , the portable hull of C, is the smallest set formed by intersecting halfspaces that contain C and are supported at points in C, and, at the same time, the largest set on which the normal cone NC can be extended monotonically. Proposition 5 Let X be a LCS. For every C ⊂ X, NC # |C = NC , Graph NC ⊂ Graph NC # , NC # ∈ M(X), and C ## := (C # )# = C # . Proof. The inclusion Graph NC ⊂ Graph NC # follows directly from the definition of C # while Graph(NC # |C ) ⊂ Graph NC is plain. To conclude it suffices to prove that C ## ⊂ C # . To this end, note that, for every x ∈ C ## and for every (a, a∗ ) ∈ NC # , we have x − a, a∗ ≤ 0. In particular, for (a, a∗ ) ∈ NC , we get x ∈ C # . Remark 6 Every (maximal) monotone extension of NC has the domain contained in C # , that is, NC ⊂ T ∈ M(X) ⇒ D(T ) ⊂ D(NC+ ) = C # . Therefore NC # is a maximal monotone extension of NC with the largest possible domain. In general, NC # is not the only maximal monotone extension of NC (even with the largest possible domain or with a normal cone structure). For example, for C = (0, 1] ⊂ R, NC admits NC and NC # as two different maximal monotone extensions; moreover NC has an infinity of maximal monotone extensions with the largest possible domain C # = (−∞, 1]. The maximality of the subdifferential allows us to reprove and extend some of the Bishop-Phelps results (see [1]). 5
Theorem 7 Let X be a LCS. If, for every closed convex C ⊂ X, NC ∈ M(X), then, for every closed convex C ⊂ X, Supp C is dense in bd C. Proof. If int C = ∅ then Supp C = bd C and we are done. In case int C = ∅ we have C = bd C and we prove that, for every closed convex U such that C ∩ int U = ∅, Supp C ∩ U = ∅. Let U be closed convex such that C ∩ int U = ∅. Assume, by contradiction, that NC |U (x) = {0}, for every x ∈ C ∩ U . Then M(X) NC∩U = NC + NU = NU |C ⊂ NU ∈ M(X), which provides the contradiction U ⊂ C, i.e., int C = ∅. Remark 8 The previous results still holds for a fixed closed convex C ⊂ X if, for every U ⊂ X closed convex such that C ∩ int U = ∅, NC∩U ∈ M(X). Theorem 9 Let X be a LCS and let C ⊂ X be closed convex, free of lines, and finitedimensional, that is, dim(span C) < ∞. Then NC ∈ M(X) and clw∗ R(NC ) = clw∗ (dom σC ). If, in addition, C is bounded, then clw∗ R(NC ) = X ∗ . Proof. For every x ∈ C the set [σC−x < 0] is non-empty. Under the given assumptions σC is w∗ −continuous on intw∗ (dom σC ) and intw∗ [σC−x < 0] = ∅. For every x∗ ∈ clw∗ (dom σC ) and every weak-star closed convex neighborhood V of x∗ , σC + ιV ∈ Γw∗ (X ∗ ), and σC + ιV is w∗ −continuous on intw∗ (dom σC ∩ V ); whence ∂(σC + ιV ) = ∂σC + NV and Graph(∂(σC + ιV )) = ∅. That yields V ∩ D(∂σC ) = ∅ and, as a consequence, clw∗ R(NC ) = clw∗ (dom σC ). From [σC−x < 0] ⊂ dom σC ⊂ cl w∗ R(NC ) and intw∗ [σC−x < 0] = ∅ we know that [σC−x < 0] ∩ R(NC ) = ∅. According to Theorem 3, NC ∈ M(X). If, in addition, C is bounded, then dom σC = X ∗ . Proposition 10 Let X be a LCS. If, for every f ∈ Γ(X), Graph ∂f = ∅, then for every f ∈ Γ(X), D(∂f ) is dense in dom f and R(∂f ) is weak-star dense in dom f ∗ . Proof. For every x ∈ dom f and every closed convex neighborhood V of x, f + ιV ∈ Γ(X); whence Graph ∂(f + ιV ) = ∅. Since ιV is continuous at x ∈ dom f , ∂(f + ιV ) = ∂f + ∂ιV ; in particular V ∩ D(∂f ) = ∅. Corollary 11 Let X be a LCS such that for every f ∈ Γ(X), Graph ∂f = ∅. Then for every closed convex bounded C ⊂ X the support functionals of C are weak-star dense in X ∗. Corollary 12 Let (X, · ) be a Banach space. Then for every closed convex bounded C ⊂ X the support functionals of C are strongly dense in X ∗ . If C ⊂ X is merely closed and convex then the support functionals of C are strongly dense in the domain of σC . In other words for every > 0, f ∈ X ∗ such that supC f < +∞ there is g ∈ X ∗ , x0 ∈ C such that g(x0 ) = supC g (that is, g ∈ NC (x0 )) and f − g ≤ . 6
Proof. Since X ∗ is a Banach space, for every f ∈ Γ(X ∗ ), ∂f ∈ M(X ∗ ); in particular Graph ∂f = ∅. According to our previous proposition D(∂σC ) is strongly dense in dom σC . Here “∂” denotes the subdifferential of functions defined in X ∗ with respect to the duality (X ∗ , X ∗∗ ). Note that the set of support functionals represent D(∂w∗ σC ) where “∂w∗ ” denotes the subdifferential of functions defined in X ∗ with respect to the duality (X ∗ , X). It suffices to show that D(∂w∗ σC ) is strongly dense in D(∂σC ) fact already observed in [3, Proposition 1, p. 211]. Again, if in addition, C is closed then dom σC = X ∗ .
3
Partial portable hulls and sum representability
Let (X, τ ) be a LCS and C, S ⊂ X. We denote by CS# the partial portable hull of C on S which is the intersection of all the (supporting) half-spaces that contain C and are supported at points in S, CS# := {x ∈ X | ∀a ∈ C ∩ S, a∗ ∈ NC (a), x − a, a∗ ≤ 0}. Equivalently x ∈ CS# ⇔ ∀a ∈ S ∩ Supp C, a∗ ∈ NC (a), x − a, a∗ ≤ 0, with the remarks that CS# = X when S ∩ Supp C = ∅ and that, for every C, S ⊂ X, # # CS# = ∅, C ⊂ clτ conv C ⊂ C # ⊂ CS# , CS# = CS∩C = CS∩Supp C. # # Note also that CS = C whenever S ⊃ Supp C.
C
CS#
S ∩ Supp C
A more comprehensive definition of the portable hull of a set C ⊂ X can be given with respect to G ⊂ Graph(NC ) by CG# := {x ∈ X | ∀(a, a∗ ) ∈ G, x − a, a∗ ≤ 0}, # with the remark that CS# = CGraph(N . C |S )
7
(4)
Proposition 13 Let X be a LCS and let C, S ⊂ X. Then (i) Graph(NC |S ) ⊂ Graph NC # and S
NC |S = NC # |S iff S ∩ C = S ∩ CS# . S
In particular NC # |C = NC . # # # (ii) (CS# )# S = (CS ) = CS . In particular NC # ∈ M(X). S
Proof. (i) For every a ∈ S ∩ C, a∗ ∈ NC (a), x ∈ CS# , x − a, a∗ ≤ 0, that is a∗ ∈ NC # (a); S whence NC |S ⊂ NC # |S . For the second part it suffices to prove the converse implication. S
Let x ∈ S ∩ CS# = S ∩ C, x∗ ∈ NC # (x), that is, for every y ∈ CS# , y − x, x∗ ≤ 0. For S y ∈ C we find x∗ ∈ NC (x). In particular, for S = C we have C ∩ C # = C so NC # |C = NC . ∗ (ii) For every x ∈ (CS# )# (S) and for every (a, a ) ∈ Graph(NC # ), (a ∈ S), we have S
x − a, a∗ ≤ 0. In particular, for (a, a∗ ) ∈ Graph(NC |S ), we get x ∈ CS# . We proved that # (CS# )# (S) ⊂ CS . The converse inclusion is immediate. Theorem 14 Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X. Then ϕT +NC = ϕT |C + ιC #
D(T )
×X ∗ .
(5)
Proof. According to its definition ϕT +NC (x, x∗ ) = sup{x − a, a∗ + a, x∗ + x − a, n∗ | a ∈ D(T ) ∩ C, a∗ ∈ T (a), n∗ ∈ NC (a)}, (x, x∗ ) ∈ X × X ∗ . Since the values of the normal cone are cones this yields that, for every x ∈ PrX (dom ϕT +NC ), a ∈ D(T ) ∩ C, n∗ ∈ NC (a), # we have x − a, n∗ ≤ 0, that is, PrX (dom ϕT +NC ) ⊂ CD(T ) ; from which one obtains, after ∗ taking n = 0, that ∀(x, x∗ ) ∈ X × X ∗ , ϕT +NC (x, x∗ ) ≥ ϕT |C (x, x∗ ) + ιC # (x). D(T )
The converse inequality is plain. Recall the following notion (see Definition 14 in [6]) Definition 15 Let (X, τ ) be a LCS. An operator T : X ⇒ X ∗ is representable in C ⊂ X or C−representable if C ∩ D(T ) = ∅ and there is h ∈ Γτ ×w∗ (X × X ∗ ) such that h ≥ c and [h = c] ∩ C × X ∗ = Graph(T |C ). Using the partial portable hull we can recover the representability of the sum between a representable operator and the normal cone (see Theorem 35 in [6]) Theorem 16 Let X be a LCS, let T : X ⇒ X ∗ , and let C ⊂ X be closed convex such that D(T ) ∩ int C = ∅. Then ψT +NC = ψT |C 2 σC # ×{0} or D(T )
∗
∗
∗
ψT +NC (x, x ) = min{ψT |C (x, x − u ) + σC # (u∗ ) | u∗ ∈ X ∗ }, (x, x∗ ) ∈ X × X ∗ , D(T )
[ψT +NC = c] = [ψT |C = c] + NC . In particular, if, in addition, T is C−representable then T + NC is representable. 8
(6) (7)
Proof. The identity in (6) follows from ϕT +NC = ϕT |C + ιC #
D(T )
ιC #
D(T )
×X ∗
×X ∗
and the fact that
# ∗ is continuous on int CD(T ) × X (see e.g. [7, Theorem 2.8.7(iii)]).
For (7) note first that dom ψT +NC ⊂ C × X ∗ . Hence (x, x∗ ) ∈ [ψT +NC = c] iff there is u ∈ X ∗ such that ψT |C (x, x∗ − u∗ ) + σC # (u∗ ) = x, x∗ iff (x, x∗ − u∗ ) ∈ [ψT |C = c] and ∗
(x, u∗ ) ∈ NC # .
D(T )
D(T )
If, in addition, T is (C−)representable then, using Theorem 16 in [6] and (7), we get [ψT +NC = c] = T + NC , that is, T + NC is representable.
4
Concluding remarks
Let (X, τ ) be a LCS and let us call a set C ⊂ X portable if C = C # (or NC ∈ M(X)). Some of the results of this paper can be summarized as follows: • C ⊂ X is portable iff for every x ∈ X \ C there is n∗ ∈ D(∂σC ) = R(NC ) such that x, n∗ > σC (n∗ ) or, equivalently, [σC−x < 0] ∩ R(NC ) = ∅. • If C ⊂ X is closed convex and X is a Banach space or int C = ∅, or C is weakly compact then C is portable. • For every C, S ⊂ X, CS# is portable. • If every closed convex C ⊂ X is portable then for every closed convex C ⊂ X, Supp C is dense in bd C. • Every closed convex, free of lines, and finite-dimensional set is portable. The results in this article are an incentive for studying the following problems: (P1) Find characterizations of all LCS’s X with the property that, for every closed convex C ⊂ X, NC ∈ M(X). (P2) Find characterizations of all LCS’s X with the property that, for every f ∈ Γ(X), Graph ∂f = ∅. (P3) Find characterizations of all LCS’s X with the property that, for every f ∈ Γ(X), ∂f ∈ M(X).
References [1] Errett Bishop and R. R. Phelps. The support functionals of a convex set. In Proc. Sympos. Pure Math., Vol. VII, pages 27–35. Amer. Math. Soc., Providence, R.I., 1963.
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[2] Simon Fitzpatrick. Representing monotone operators by convex functions. In Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), volume 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 59–65. Austral. Nat. Univ., Canberra, 1988. [3] R. T. Rockafellar. On the maximal monotonicity of subdifferential mappings. Pacific J. Math., 33:209–216, 1970. [4] M. D. Voisei. A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces. Math. Sci. Res. J., 10(2):36–41, 2006. [5] M. D. Voisei. The sum and chain rules for maximal monotone operators. Set-Valued Anal., 16(4):461–476, 2008. [6] M. D. Voisei. Location, Identification, and Representability of Monotone Operators in Locally Convex Spaces. Set-Valued Var. Anal., 27(1):151–168, 2019. [7] C. Zălinescu. Convex analysis in general vector spaces. World Scientific Publishing Co. Inc., River Edge, NJ, 2002.
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