Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S X its closed uit ball ad its uit sphere, respectively. If the topology o X is ot specified, the topological otios are cosidered i the orm topology. Theorem 1.1. For a Baach space X, the followig assertios are equivalet: (i) X is reflexive; (ii) B X is weakly compact; (iii) every elemet of X attais its orm. About the proof. The implicatios (i) (ii) (iii) are easy ad elemetary (they oly require kowledge of the Baach Alaoglu ad the Goldstie theorems). O the other had, the remaiig implicatio (iii) (i), kow as the James theorem, is a deep ad difficult result by R.C. James. From the history. R.C. James obtaied the separable versio of the above implicatio (iii) (i) i 1957 [A. Math. 66 (1957), 159 169], ad the geeral versio i 1963 [Studia Math. 23 (1963/1964), 205 216]. Fially, i 1964 [Tras. Amer. Math. Soc. 113 (1964) 129 140], he provided the followig geeral result. Theorem 1.2. Let C be a bouded closed covex set i a Baach space X. The C is weakly compact if ad oly if every elemet of X attais a maximum o C. The proof of this last theorem is quite ivolved. A accessible (though by o meas simple) proof, eve i a more geeral settig of complete locally covex topological vector spaces, was give by J.D. Pryce i 1966 [Proc. Amer. Math. Soc. 17 (1966), 148 155]. I last two decades, simpler proofs of the separable versio of Theorem 1.2 appeared. We preset here the proof by W.B. Moors, published i W.B. Moors, A elemetary proof of James characterizatio of weak compactess, Bull. Aust. Math. Soc. 84 (2011), 98 102. For a differet approach by usig a Simos lemma, see the paragraph 3.11.8 i the book 1
2 Fabia, Habala, Hájek, Motesios ad Zizler, Baach space theory. The basis for liear ad oliear aalysis, CMS Books i Mathematics/Ouvrages de Mathématiques de la SMC, Spriger, New York, 2011. 2. Boudaries Defiitio 2.1. Let X be a ormed space, ad K X a bouded set. A set B K is a boudary or (James boudary) for K if every x X attais its supremum over K at some poit of B, that is, x X b B : b (x) = sup x (x). x K Examples 2.2. Let K X be a w -compact set (where X is a ormed space). (a) K is a boudary for itself. (b) K is a boudary for K. (c) If K is also covex, the ext K is a boudary for K by Bauer s maximum priciple (which is a easy cosequece of the Krei Milma theorem). The followig simple but importat fact is left to the reader as a easy exercise o the Hah Baach theorem. Exercise 2.3. Let K X be a w -compact covex set, ad B a boudary for K. The K = cov w B. Let us start with a simple lemma of topological ature. By U(0) we deote the family of all eighborhoods of 0 (i a topological vector space). It is a well-kow fact that every elemet of U(0) cotais a closed oe. Lemma 2.4. I a Hausdorff topological vector space, let S, K ad K ( N) be closed sets such that: (a) K is compact, (b) S K =, (c) S N K, ad (d) for every V U(0), K K + V for each sufficietly large. The there exists M N such that S M K. Proof. By (a),(b), there exists a closed V U(0) such that S (K + V ) =. Notice that K + V is closed. By (d), there is M N such that >M K K + V. It follows that K K + V. >M
3 Usig the geeral equality A B = A B, we obtai that S K = K K, N M >M where the last set is disjoit from S. Thus S M K. Remark 2.5. Let us recall the followig two results which will be eeded i the sequel. (a) Let K be a compact covex set i a locally covex Hausdorff topological vector space, ad A K. The followig assertios are equivalet: (i) K = cov A ; (ii) ext K A. (The implicatio (ii) (i) is the Krei Milma theorem, while the other oe is its coverse due to Milma.) (b) Let X be a Baach space, ad D X a w -closed covex set with oempty iterior. The the w -support poits of D are dese i D. (This dual versio of the Bishop Phelps theorem ca be deduced from the stadard Bishop Phelps theorem applied to the pre-polar D X. Let us remark that Phelps proved a stroger result without the assumptio about oempty iterior, but his proof follows a differet approach.) Now we are ready for the basic theorem of this sectio. Theorem 2.6. Let X be a Baach space, ad K X a w -compact covex set. If {C } is a sequece of w -compact covex sets whose uio cotais a boudary B for K, the ( ) K cov C (closure i the orm topology!). Proof. We ca (ad do) suppose that 0 B ad C K for each. Fix a arbitrary ε > 0, ad cosider the sets N K := C + ε B X ( N), D := covw ( N K ). Notice that the sets K ad D are w -compact sice they are cotaied i K + εb X. Moreover, B N C N it K, ad hece 0 B it D. Let x D be a w -support poit of D, that is, there exists x X \ {0} such that x(x ) = max x(d) = 1. Cosider the face F := [x = 1] D of D. Assume for the momet that F K. Sice K D, we must have max x(k) = 1. Take b B [x = 1]. But we already kow that b it D, which cotradicts the fact that b F D.
4 Sice F is a extremal set for D, Re- Thus we must have F K =. mark 2.5(a) implies that ext F ext D N K w. Put S := F N K w, ad otice that ext F S N K w ad S K =. Moreover, if V X is a w -eighborhood of 0 the ε B N X V for some N N, ad hece >N K K + ε B N X K + V. By Lemma 2.4, there exists M N such that ext F S K. By the Krei Milma theorem, x F = cov w (ext F ) cov ( M w K ) = cov ( M K ) ( cov N C ) + εbx. This shows that all w -support poits of D are cotaied i cov ( N C ) + εb X. By Remark 2.5(b), D cov ( N C ) + 2εBX. Cosequetly, ( ) K D cov C + 2εB X, N ad we are doe sice ε > 0 was arbitrary. M 3. Applicatios: theorems by Raiwater, Rodé, ad James Corollary 3.1. Let X be a Baach space, K X a w -compact covex set with a boudary B K. Let {x } X be a bouded sequece, ad x X such that b (x ) b (x) for each b B. The x (x ) x (x) for each x K. Proof. For simplicity, deote y := x x. Fix a arbitrary ε > 0, ad cosider the sets C := K k [ yk ε ] ( N). These sets form a odecreasig sequece of w -compact covex sets. Notice that B C. By Theorem 2.6, the set C := C = cov ( C ) is orm-dese i K. Moreover, lim sup y (c ) ε for each c C. Sice C is dese i K, ad the sequece {y } is bouded, we easily obtai that lim sup y (x ) ε for each x K,
ad we are doe by arbitrariety of ε > 0. The last corollary immediately implies the followig useful criterio of weak covergece of a bouded sequece. Theorem 3.2 (Raiwater 1963). Let {x } be a bouded sequece i a Baach space X, ad x X. Assume that e (x ) e (x) for each e ext B X. The x x i the weak topology. Proof. Apply Corollary 3.1 to K = B X ad B = ext B X. The followig theorem was first proved by Rodé (1981) by a completely differet method for w -compact covex sets. The versio stated here was proved by Godefroy (1987) by usig a lemma by Simos. The preset proof via Theorem 2.6 seems to be the most elemetary kow oe. Theorem 3.3 (Rodé s theorem (or Rodé Godefroy theorem)). Let X be a Baach space, C X a closed, covex ad bouded set with a separable boudary B C. The C = cov B, ad the set C is w -compact. Proof. Defie K = C w. The K is a w -compact covex set ad B is a boudary for K. Let {b } B be a dese sequece. Give a arbitrary ε > 0, defie C := (b + εb X ) K ( N), ad otice that B C. By Theorem 2.6, K cov C cov (B + εb X ) cov (B + εb X ) + εb X cov B + 2εB X. Sice ε > 0 was arbitrary, we have K cov B C K, ad we are doe. Corollary 3.4. If X is a Baach space such that ext B X is separable, the X (ad hece also X) is separable. Now, we are goig to prove the followig separable versio of Theorem 1.2. Theorem 3.5 (James). Let C be a separable, closed, covex ad bouded set i a Baach space X. The C is w-compact if ad oly if each elemet of X attais a maximum over C. Proof. The oly if part is obvious. To show the if part, assume that each elemet of X attais a maximum over C. This meas that C, if cosidered as a subset of X, is a separable boudary for itself. By Rodé s theorem, C is w -compact i X, ad hece w-compact i X. Corollary 3.6 (James 1957). A separable Baach space X is reflexive if ad oly if each elemet of X attais its orm. 5