PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 10, Pages 3025{3038 S 0002-9939(03)07066-7 Article electronically published on April 30, 2003 PHELPS' LEMMA, DANES'˘ DROP THEOREM AND EKELAND'S PRINCIPLE IN LOCALLY CONVEX SPACES ANDREAS H. HAMEL (Communicated by Jonathan M. Borwein) Abstract. A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equiv- alence of this theorem, Ekeland's principle and Dane˘s' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997. We show that a different formulation of Ekeland's principle in locally con- vex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version. 1. Introduction The famous 1972 Ekeland theorem called the \Variational Principle" is one of the most frequently applied results of nonlinear functional analysis. For example, the progress of variational calculus and optimal control over the last 25 years is unthinkable without Ekeland's principle. Around the same time, but independently of each other, a list of theorems was discovered in which all of the theorems characterize the completeness of the under- lying space and are equivalent to Ekeland's principle in metric spaces, namely the Krasnosel'skii-Zabrjeko theorem on normal solvability of operator equations [23], the Kirk-Caristi fixed point theorem [4], and the Dane˘sdroptheorem[6]. Over the last three decades, a great deal of effort has gone into looking for another equivalent formulation or generalization of Ekeland's principle; see e.g. [18], [10], [16], [17]. In this paper we continue this effort by showing that a 1974 result of Phelps (Lemma 1.2 of [20] with precursor Lemma 1 in [2], called Phelps' lemma in the following) can be generalized to sequentially closed sets of locally convex spaces, that the locally convex variant can be proven using only the corresponding Banach space version and that this procedure is applicable for proving Ekeland's principle and the drop theorem in locally convex spaces. Received by the editors May 17, 2001. 2000 Mathematics Subject Classification. Primary 49J40, 46A03. Key words and phrases. Phelps' lemma, Ekeland's variational principle, Dane˘s' drop theorem, efficiency, locally convex space. c 2003 American Mathematical Society 3025 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3026 ANDREAS H. HAMEL Additionally we show the equivalence of these theorems with each other, sur- prisingly to their Banach space counterparts and to a second variant of Ekeland's principle due to Fang [9] specialized to locally convex spaces. Finally, solving a problem stated by G. Isac in [14], we point out the equivalence of the drop theorem in locally convex spaces to one of Isac's theorems which relates Ekeland's principle to Pareto efficiency. We close the paper with some conclusions concerning the relationships of different versions of the represented theorems. The appendix contains the Banach space versions of Phelps' lemma, Ekeland's principle and Dane˘s' drop theorem as well as a separation theorem. 2. Preparatory results Let us start with some notational arrangements. Let X be a real vector space. A functional p : X ! (−∞; +1) is called a seminorm if it satisfies (i) p (αx)=jαj p (x) for all x 2 X, α 2 (−∞; +1), (ii) p (x + y) ≤ p (x)+p (y) for all x; y 2 X. f g Following Treves [22] a pair X; pi i2I is said to be a locally convex topolog- f g ical vector space (short: locally convex space) if pi i2I is a family of seminorms satisfying (iii) i 2 I, α>0 implies the existence of j 2 I such that pj = αpi, (iv) i 2 I, q is a seminorm such that q (x) ≤ pi (x) for every x 2 X implies the existence of j 2 I such that pj = q, 2 2 f g (v) i1;i2 I implies the existence of j I such that pj =sup pi1 ;pi2 . The locally convex space X is said to be Hausdorff (or separated) if (vi) x; y 2 X and x =6 y implies the existence of j 2 I such that pj (x) =6 pj (y). f g The family pi i2I of seminorms satisfying (iii) - (v) is called the spectrum of X and is denoted by spec (X). The set spec (X) generates a topology on X such that addition and multiplication by scalars are continous operations and p 2 spec (X) f g if and only if p is uniformly continuous on X. A subfamily pλ λ2Λ of spec (X) is said to be a base of continuous seminorms if for every p 2 spec (X)onX there are α>0andpλ, λ 2 Λ, such that p (x) ≤ αpλ (x) for every x 2 X. A Hausdorff locally convex topological vector space always has a base of continuous seminorms. f g It can be constructed in the following way. Starting with spec (X)= pi i2I we define Λ = fλ ⊂ I;λ finiteg and pλ (x)=suppj (x)forx 2 X; λ 2 Λ: j2λ As usual, we define the Minkowski functional of a set A ⊂ X to be µA (x):=inffα>0: x 2 αAg : The main idea of our proofs is to reduce the situation to the Banach space case. We present some preparatory results containing the core of the method. Proposition 1. Let X be a locally convex Hausdorff space and T ⊂ X a convex balanced and bounded set. Then the Minkowski functional µT (·) defines a norm on the linear space span T . Proof. Since µT is positive homogeneous and T balanced we may conclude µT (αx) = jαj µT (x) for all α 2 R and x 2 span T . The convexity of T implies the convexity of µT . This fact, together with the positive homogeneity of µT , implies the validity License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PHELPS' LEMMA AND DANES'˘ DROP THEOREM 3027 of the triangle inequality for µT . Obviously, µT (0) = 0. It remains to show that µT (x) = 0 implies x =0.SinceT is bounded in X, for every neighborhood U of 2 1 ⊂ 2 1 0 X there exists σ>0 such that σ T U.SinceµT (x)=0wehavex σ T for each σ>0, hence x 2 U for every neighborhood U of 0 2 X which implies x =0. If the assumptions of Proposition 1 are satisfied we denote the normed space k·k (span T; µT )by(XT ; T ). Proposition 2. Let X be a locally convex Hausdorff space and T ⊂ X asequentially k·k complete convex balanced and bounded subset of X.Then(XT ; T ) is a Banach space. k·k Proof. It remains to show that (XT ; T ) is complete with respect to the topology k·k induced by T . k·k f }⊂ First, we show that T is complete with respect to T .Let xn T be a k·k Cauchy sequence with respect to T . Then it is Cauchy with respect to the locally convex topology: Because of the boundedness of T for every i 2 I there exists αi > 0 such that T ⊂ αiUi where Ui := fx 2 X : pi (x) ≤ 1g and 1 − 1 − ≤ − pi (xm xn)= µUi (xm xn) µT (xm xn) : αi αi Hence there exists x 2 X such that xn ! x in X.Moreover,x 2 T since T is ! k·k sequentially closed in X.Next,weshallshowxn x with respect to T .Fix ">0. Since fxng is Cauchy in XT we have xm − xn 2 "T for all m; n sufficiently large. Letting n !1we conclude xm − x 2 "T for all m sufficiently large since T is sequentially closed in X. This implies − k − k ≤ µT (xm x)= xm x T " ! k·k for all m sufficiently large which gives xn x with respect to T . Finally, we show that an arbitrary Cauchy sequence fxn}⊂XT has a limit point 2 k·k fk k }⊂R y XT with respect to T .Notethat xn T is Cauchy since jk k −k k |≤k − k xn T xm T xn xm T ; 2 R k k ≥ nhence convergento to some α . Assuming xn T 1 for all n we claim that xn k k is Cauchy in T with respect to k·k . This follows from xn T T xn xm xn xn xn xm − = − + − kx k kx k kx k kx k kx k kx k n T m T T n T m T m T m T T 1 1 1 ≤ − kx k + kx − x k kx k kx k n T kx k n m T n T m T m T kx k ≤ − n T k − k 1 k k + xn xm T xm T 1 ≤ (kx k −kx k )+kx − x k k k m T n T n m T xm T ≤ k − k 2 xn xm T : License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3028 ANDREAS H. HAMEL xn Because T is complete we conclude that k k ! z 2 T . Moreover, we have xn T ! 2 k·k xn αz XT with respect to T because k − k ≤k −k k k k − k xn αz T (xn xn T z)+( xn T z αz) T ≤k(x −kx k z)k + k(kx k z − αz)k n n T T n T T x ≤kx k · n − z +(kx k − α) ·kzk : n T k k n T T xn T T k k ! Since xn T α, the very right-hand side of the last inequality chain tends to zero, which completes the proof of the proposition.