DEMONSTRATIO MATHEMATICA Vol. XLI No 1 2008
S. Lahrech, A. Jaddar, J. Hlal, A. Ouahab, A. Mbarki
ON THE PRINCIPLE OF UNIFORM BOUNDEDNESS FOR LSC CONVEX PROCESSES IN A STRICTLY N LOCALLY CONVEX SPACES
Abstract. In this paper we establish the principle of uniform boundedness for LSC convex processes in some class of locally convex spaces (strictly Af locally convex spaces). Thus, we generalize the same result established by S. Lahrech in [1] for sequentially con- tinuous linear operators.
1. Introduction Convex processes were introduced by Rockafellar ([2], [3]). These axe set-valued maps whose graphs are convex cones. As such, they provide a powerful unifying formulation for the study of linear bounded operators, convex cones, and linear programming. The purpose of this paper is to establish the principle of uniform bound- edness for LSC convex processes in some class of locally convex spaces (strictly M locally convex spaces). Throughout this section, (X, A) and (Y, fi) are two locally convex spaces. An operator T : (X, A) —> (Y, fi) is said to be (A, ^-sequentially con- tinuous if for every sequence (xn) of X and every x £ X such that xn—>x one has Txn —> Tx. T is said to be (A, /x)-bounded if T sends A-bounded sets into n bounded sets. Clearly, continuous operators are sequentially con- tinuous and sequentially continuous operators are bounded, but in general, converse implications fail. Let X'x, and X\ denote the families of con- tinuous linear functionals, sequentially continuous linear functionals, and bounded linear functionals on (X, A) respectively. In general, the inclusions X[ C c X^ are strict. Recall that a subset B of X is said to be condi-
Key words and phrases: LSC convex processes, strictly N locally convex spaces, principle of uniform boundedness. 1991 Mathematics Subject Classification: 49J52, 49J50, 47B37, 46A45. 160 S. Lahrech, A. Jaddar, J. Hlal, A. Ouahab, A. Mbarki
tionally sequentially compact in (X, A) if for every sequence (xn) of B, there exists a subsequence (xnk) of (xn) converging in (X, A). Let B{X\) and C{X\) denote respectively the families of bounded sets and conditionally sequentially compact sets in (X, A). Let a C B(X\). By (A) we denote the locally convex topology on X^ generated by the family of semi norms
PciT) = sup|Tx|, Cea. xec Let (3(X,a) denote the topology on X of uniform convergence on Co-(A)- bounded sets of Xy A locally convex space (X, A) is said to be strictly A/*-locally convex, if there exists a complete norm || • ||x on X, a family a of subsets of X such that
C(Xx) CaC B(X) = B(X, || • \\x) C B{Xx) and (J C = X. Cea We can give many examples of such spaces. For example, if we take any Banach space (X, || • ||x), then it is a strictly A/"-locally convex space. The weak topology a(X, X'), where X is a Banach space and X' is the space of continuous linear functionals on (X, || • ||x) is also a strictly AMo- cally convex topology. Assume now that (X, A) and (Y, ¡i) are two strictly AMocally convex spaces. Then, there exists || • ||x, || • ||y, c, o\ such that
C{Xx) C a C B{X) C B(Xx) and C(FM) C C B{Y) C
Moreover, |JCeaC = X and Uceo-i C = Y. Denote by Bi(Y) the unit ball in (y, || • ||y). Let X' be the space of continuous linear functionals on
(X, || • ||jr). Put Fl = B(Y£,Cn (/*)), where B(Yj, (//)) denote the class of bounded sets in Y* with respect to the locally convex topology (p)). S. Lahrech has proved in [1] that X^ c X' and
B(Xi,C,W)cB(X',\\-\\x>). Therefore,
VA 6 Fi My e Y, PA(y) = sup \f(y)\ < +oo. /eA Thus, the locally convex topology (3(fi,ai) is generated by the family {Pa}a<=Fi of semi norms on Y.
According to [1], a sequence (Tn), Tn : X —> Y of linear operators is said to be (|| • ||x, H, cri)-bounded if
VILGB(Y%W) 3D >0VXG51(X) V/G AVneN \f{Tnx)\ It has been shown in [1] that if (X, A, || • Hx,®-), (Y,n, II • lly^i) are two strictly Af-locally convex spaces, and if Tn : X —» Y is a sequence of (A, //)- sequentially continuous linear operators satisfying the condition: for each x G X the sequence (Tnx)n is bounded in (Y,/?(/z,o"i)), then the sequence 2. Principle of uniform boundedness for LSC convex processes in a strictly ^/-locally convex spaces Let $ : X —> Y be a multifunction. The domain of $ is the set £>($) = {xeX : $(x) + 0}. We say $ has nonempty images if its domain is X. For any subset C of X we write (C) for the image (JxeC and the range of $ is the set /?(<£) = We say $ is surjective if its range is Y. The graph of $ is the set G($) = {(x,y)£XxY:ye(x)}. We say $ is a process if its graph is a cone. $ is said to be convex, or closed if its graph G($) is likewise. For example, we can interpret linear closed operators as closed convex processes. A convex process $ : X —> Y is said to be (A, ¡i) -bounded if it maps every bounded set of (X, A) into bounded set in (Y, /¿). Assume that (X, A, || • ||x, c) and (Y, fi, || • ||y, o\) are two strictly AMocally convex spaces. Recall that a convex process $ : X —> Y satisfying the condition: Vrr0 € X, $(x0) C (A,fi) - limsup$(a;) = {y G Y • v Xn • £0, x—>xo 3(z„fc) subsequence of (xn) 3yk & $(xnk) such that y^ ^ y} is called a (A, /x)-LSC convex process. A sequence : X —> Y of convex processes is said to be (|| • ||x, A4,0"i)- bounded if V^ G B{Y*,U{»)) 3c! > 0 Vx 6 B^X) VfeAVne NVy G $n(x) \f{y)\ < Cl. Now we can give the main result of our paper. THEOREM 1. Let (X, A, || • ||x,cr) and (Y,fi, || • ||y,oi) be two strictly N- locally convex spaces and let : X —• Y be a sequence of (A, p)-LSC convex processes such that the set Un>i ^n(^) is bounded in (F,/3(/i,cri)) for each x G X. Then (3>n) is (|| • \\x, -bounded. 162 S. Lahrech, A. Jaddar, J. Hlal, A. Ouahab, A. Mbarki Proof. Let A G 1(n)). For any integer s > 1 put Xs = {xeX: V/ G A Vn G N Vy G $n{x) |/(y)| < s}. Since Un>j $n(cc) is bounded in (Y, /?(//, ai)), then X = On the other hand, Vs > 1 Xs is closed in (X, || • )• Indeed, Fix s > 1 such that Xs 0 and let (xr) be a sequence of Xs converging to some x G X with respect to the norm || • ||x • We have xr G Xs (Vr). Therefore, V/G-AVnGiVVt/G $n(x) |/(y)| < s. Let / G A, n G N, y G $n(x). Hence, without loss of generality, we can assume that there is yr G $n(xr) converging to y with respect to the topology fi. Hence, using the fact that / G Y*, we deduce f(yr) f(y)- Consequently, since \f(yr)\ < s (Vr), we conclude that \f(y)\ < s. This implies that x G Xs. Thus, Xs is closed in (X, || • ||x). Applying Baire theorem, we deduce that int XSo / 0 for some so > 1. Taking XQ € X, r > 0 such that XQ + rB\(X) c XSQ and observing that for each n, is a convex cone, we deduce V/GiVn>lVze Bx{X) y y G $„(*)\f(y)\ < r Thus, we achieve the proof. REMARK 2. Let us remark that if Tn : X —> Y is a sequence of (A, /x)- sequentially continuous linear operators satisfying the condition: for each x G X, the sequence (Tnx)n is bounded in (F, /?(//, References [1] S. Lahrech, On the principle of uniform boundedness in a strictly N-locally convex space, Int. J. Pure Appl. Math. 16, No. 2 (2004), 141-145. [2] R. T. Rockafellar, Monotone processes of convex and concave type, Mem. Amer. Math. Soc., 1967. No. 77. [3] R. T. Rockafellar Convex Analysis, Princeton University Press, Princeton, N.J., 1970. [4] J. M. Borwein, A. S. Lewis, Convex Analysis and Nonlinear Optimization, CMS Books in Mathematics. Gargnano, Italy, September 1999. [5] A. Wilansky, Modern Methods in TVS, McGraw-Hill, 1978. [6] G. Ko the, Toplogical Vector Spaces, I, Springer-Verlag, 1983. [7] Haim Brezis, Analyse fonctionnelle,Théorie et applications, Masson, Paris, 1983. On the principle of uniform boundedness S. Lahrech, J. Hlal, A. Ouahab DEPT OF MATHEMATICS FACULTY OF SCIENCE MOHAMED FIRST UNIVERSITY OUJDA, MOROCCO (Dynamical system and Optimization Group, GAFO Laboratory) E-mail: Lahrech, Hlal, [email protected] A. Jaddar NATIONAL SCHOOL OF MANAGMENT OUJDA, MOROCCO (Dynamical system and optimization group, gafo laboratory) E-mail: [email protected] A. Mbarki NATIONAL SCHOOL OF APPLIED SCIENCES OUJDA, MOROCCO (Dynamical system and Optimization Group, GAFO Laboratory) E-mail: [email protected] Received November 18, 2006; revised version June 1, 2007.