Gruenhage compacta and strictly convex dual norms

GR UENHA GE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS RICHARD J. SMITH Abstract. W e prov e t hat if K is a Gruenhage co mpact space then C ( K ) ∗ admits an equ iv alen t, strictly con v ex dual n orm. As a c orollary , w e sho w that if X is a Banac h space and X ∗ = span |||·||| ( K ), where K is a Gruenhage compact in the w ∗ -topology and ||| · ||| is equiv alen t to a coarser, w ∗ -low er semicon tinu ous norm on X ∗ , then X ∗ admits an equiv alent, strictly con vex dual norm. W e give a partial con verse to the first result by sho wing that i f Υ is a tree, then C 0 (Υ) ∗ admits an equiv alent , strictly conv ex du al norm if and only if Υ is a Gruenhage space. Finally , we presen t some stability prop erties satisfied b y Gruenhag e spaces; in particular, Gruenhage spaces are stable under perf ect images. 1. Introduction and preliminaries In renorming theory , w e determine the exten t to which the norm of a given B a- nach space can b e mo dified, in order to improve the geo metry of the cor resp onding unit ball. Naturally , the structura l theory of Banach spaces plays an impo rtant par t in t his field but, in recen t times, there has b een a mo ve tow ard a more non-linear, top ological approach. This new outlo ok led to the so lution of some long-s tanding problems, as well as pro ducing some completely unexpected results. Recall that a norm || · || o n a real Banach spac e X is called strictly c onvex , or r otund , if || x || = || y || = 1 2 || x + y || implies x = y . W e say that || · || is lo c al ly uniformly r otu nd , or LUR , if, g iven a p oint x a nd a sequence ( x n ) in the unit s phere S X satisfying || x + x n || → 2, we ha ve x n → x in norm. If || · || is a dual norm o n X ∗ then || · || is called w ∗ -LUR if, given x and ( x n ) a s a bove, we hav e x n → x in the w ∗ -top ology . F or a dual norm, evidently LUR ⇒ w ∗ -LUR ⇒ strictly conv ex. It turns out that, in some contexts, these ostensibly conv ex , geometrica l prop er- ties of the no rm can be ch arac ter ised relatively simply in purely no n-linear, top o- logical terms. Given a compact, Hausdorff space K , we denote the Banach s pace of contin uous real-v a lued functions on K by C ( K ), and identif y C ( K ) ∗ with the space of regular , sig ned Borel measures on K . Ra ja prov ed that if K is a co m- pact space then C ( K ) ∗ admits an e quiv alent, dual L UR norm if and o nly if K is σ -discr ete [8]; that is, K is a countable union o f sets, e ach o f which is discrete in its s ubs pa ce to po logy . Moreover, C ( K ) ∗ admits a n eq uiv alent w ∗ -LUR no rm if and Date : October 2007. 2000 Mathematics Subje c t Classific ation. Primary 46B03; Secondary 46B26. Key wor ds and phr ases. Gruenhage space, r otund, strictly conv ex, norm, tree, renormi ng the- ory , p erfect image, contin uous image. Some of this research was conducted duri ng a visit to the Universit y of V alencia, Spain. The author is grateful to A. Molt´ o for the invitat ion and subsequen t discussions. He also wishes to thank S. T roy anski and V. Monte sinos for interesting discussi ons and remarks. 1 2 RICHARD J. SMITH only if K is descriptive [9]; the definition of a descriptive compac t spa ce is given below. Ra ja also proved that X ∗ admits an equiv alent w ∗ -LUR norm if a nd only if B X ∗ is descriptive in the w ∗ -top ology . Regarding stric tly conv ex norms, the authors of [6] recently sho wed that X , which can b e a dua l s pace, admits a n equiv alent, strictly convex, σ ( X , N )-lower semicontin uous norm if and o nly if the sq uare B 2 X has a certain linear , top olog ical decomp osition with resp ect to a given no rming subspace N ⊆ X ∗ . In this pap er, we examine wha t can be done without the linea rity , and without explicit r eference to the square. Using Gruenhage compacta, w e obtain a sufficient co ndition fo r a dual spa ce X ∗ to admit an eq uiv a lent , strictly c onv ex dua l norm. This condition cov er s all established clas ses of Banach space known to b e so renor mable, including the dua ls of all we akly c ountably determine d , or V a ˇ s´ ak, s paces. It also cov ers the more gener al class o f ‘des criptively gener ated’ dual s paces, introduced recently in [7]. W e define descriptive compact spaces and re la ted notio ns. All top olo gical spaces are assumed to be Hausdorff. A family of subsets H of a top ological space X is called isolate d if, g iven H ∈ H , ther e ex ists an open set U that includes H and misses every other element of H ; i.e. H is discrete in the union S H . The family H is ca lled a n et work for K if, g iven t ∈ U , where U is op en, there exists H ∈ H such that t ∈ H ⊆ U . In other words, a netw o rk is a bas is , but without the requirement that its elements b e op en subsets. Finally , we say that a compac t space K is descriptive if it has a net work H that is σ -isolate d ; that is , H = S n H n , where each H n is a iso lated family . The cla ss of desc r iptive compact spac es is ra ther larg e. It includes tw o cla sses of top ological s paces that hav e featured prominently in non- separable B anach space theory , namely Eb erlein and Gul’ko compacta; s ee, for exa mple [2 ]. It also in- cludes a ll σ -discrete co mpact spa ces; in particular, all compa c t K such that the Cantor deriv ative K ( ω 1 ) is empty , wher e ω 1 is the least uncountable ordina l. More information ab out de s criptive compact spaces can b e found in [7]. More genera lly , we say that a top olo gical space X is fr agmentable if there exists a metric d on K , with the pro per ty that g iven ε > 0 and non-empty E ⊆ T , there is an op en set U such that U ∩ E is non-empty and the d -diameter of E ∩ U do es not exceed ε . General fragmentable compact s pa ces are not particula rly well- behaved from the p oint of view of r enorming. Indeed, since every s cattered space is fragmented b y the discrete metric, the c ompact ω 1 + 1 is frag ment able, and it is well- known tha t C ( ω 1 + 1) ∗ do es not admit an equiv alent, strictly convex dua l no r m; see, for example [1, Theo r em VI I.5.2 ]. On the other hand, if X ∗ do es a dmit an equiv alent, strictly conv ex dual norm, then B X ∗ is fr agmentable in the w ∗ -top ology [11]. The class of Gruenhage compact spaces fits b etw een those of descriptive and fragmentable s paces. Definition 1.1 (Gruenhage [4]) . A to p olo gical space X is called a Gruenhage space if there e xist families ( U n ) n ∈ N of op en s ets such that given distinct x, y ∈ X , ther e exists n ∈ N and U ∈ U n with t wo prop erties: (1) U ∩ { x, y } is a singleton; (2) either x lies in finitely many U ′ ∈ U n or y lies in finitely many U ′ ∈ U n . If we were to follow Gruenhage’s definition to the letter, the sequence ( U n ) ab ov e would have to cover X as well, but this demand is not necessa ry as prop erty (1) GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 3 forces the seq uence to cov er a ll p oints of X , with a t most one e x ception. Gruenhage calls such s equences σ -distributively p oint-finite T 0 -sep ar ating c overs of X . In the next section, w e in vestigate the r ole of Gruenha g e spaces in r enorming theory . In the third section, we give a par tial c onv ers e to Theorem 2.6, the principal result o f the se cond sectio n, a nd, by vir tue of ex amples, get so me measur e of the gap b e tw een descriptive compact spaces a nd Gruenhage compact s paces. The last section is devoted to proving certa in stability prop erties of the c lass of Grue nha ge spaces a nd its sub class of c o mpact spa ces. 2. Gruenha ge comp a ct a and renorming W e shall say that a family H o f subsets of a top olog ic al s pace X sep ar ates p oints if, given distinct x, y ∈ X , there exists H ∈ H such that { x, y } ∩ H is a s ingleton. It sho uld b e noted that some authors demand more of p oint separation, na mely that H can b e chosen in such a wa y that { x, y } ∩ H = { x } . The next pr o po sition bring s tog e ther some equiv a lent formulations of Gruen- hage’s definition that will b e of use to us. Prop ositio n 2 .1. L et X b e a top olo gic al sp ac e. The fol lowing ar e e quivalent. (1) X is a Gru en hage sp ac e; (2) ther e exists a se qu enc e ( A n ) of close d sets and a se quenc e ( H n ) of families such that S n H n sep ar ates p oints, and furthermor e e ach element of H n is an op en subset of A n and disjoint fr om every other element of H n ; (3) ther e exists a se quenc e ( U n ) of families of op en subsets of X and sets R n , such that S n U n sep ar ates p oints and U ∩ V = R n whenever U, V ∈ U n ar e distinct. Pr o of. (1) ⇒ (2) follows directly from [15, Pro p os ition 7 .4]. Supp ose that (2) holds. T o obtain (3), s imply define R n = K \ A n and set U n = { H ∪ R n | H ∈ H n } . Finally , if (3) holds, define V n = { R n } . Given distinct x, y ∈ X , there e x ists n and U ∈ U n such that { x, y } ∩ U is a sing le to n. Let us assume that x ∈ U . There are t wo cases. If x ∈ R n then y / ∈ R n bec ause R n ⊆ U , thu s { x, y } ∩ R n is a single to n and, since V n is a singleto n, x is in exactly one element of V n . Alterna tiv ely , we assume that x / ∈ R n . Then x ∈ V ∈ U n forces V = U . Hence x is in exactly one element of U n . This shows tha t X is Gr uenhage.  The se cond fo r mu lation pr esented in the prop osition ab ov e pro mpts the following definition. Definition 2.2 . Let X b e a top ologica l spa ce. W e call ( A n , H n ) a le gitimate system if A n and ( H n ) a re a s in P rop osition 2 .1, part (2). W e s ay that H = S n H n is the union of the sy stem. The next result follows easily . Corollary 2.3. A descriptive c omp act sp ac e is Gruenhage. Pr o of. In [9], Ra ja shows that if K is a descriptive compact spa ce then there exists a leg itimate sy s tem ( A n , H n ) such that its union H is a ne tw ork for K .  W e will s p end a little time prepa ring our legitimate sys tems for battle. W e can and do ass ume for the rest of this section that every legitima te system ( A n , H n ), with union H , satisfies three pro per ties: 4 RICHARD J. SMITH (1) H is closed under the taking of finite intersections; (2) K \ A n ∈ H fo r a ll n ; (3) A n \ S H n ∈ H fo r a ll n . Indeed, we first extend the system ( A n , H n ) by adding the pairs ( K, { K \ A n } ) and ( A n \ S H n , { A n \ S H n } ) for every n . W e denote the extended system ag a in by ( A n , H n ) and then co ns ider, for each non- empt y , finite F ⊆ N , the pairs ( A F , H F ), where A F = T n ∈ F A n and H F = { T n ∈ F H n | H n ∈ H F } . A family H of pairwise disjoint subsets o f K is called sc atter e d if there exists a well-ordering ( H ξ ) ξ<λ of H such that S ξ<α H ξ is op en in S H for a ll α < λ . Equiv alently , H is sc attered if, given non-empty M ⊆ S H , there exists H ∈ H such that M ∩ H is non-empty and o pen in M . Sca ttered families naturally generalise is o lated ones. The following lemma is a simple extension o f Rudin’s res ult that Radon measures on sca ttered co mpa ct s paces are atomic. W e can sta te it in greater genera lit y than r equired, without compro mis ing the simplicity of the pro of. W e will say that H ⊆ K is universal ly R adon me asur able (uRm) if, given p ositive µ ∈ C ( K ) ∗ , there ex ist Borel sets E , F such tha t E ⊆ H ⊆ F a nd µ ( E ) = µ ( F ); equiv alently , H can b e measured by the completio n of e a ch such µ , which we again denote by µ . Lemma 2.4. If H is a sc att er e d family of uRm subsets of a c omp act sp ac e K t hen S H is uRm and µ ( S H ) = P H ∈ H µ ( H ) for every p ositive µ ∈ C ( K ) ∗ . Pr o of. T a ke a well-ordering ( H ξ ) ξ<λ of H a nd op en sets U α , α < λ , s uc h that H α ⊆ U α and U α ∩ H β is empty whenever α < β . W e pro ceed by transfinite induction on λ ; no te that by σ -a dditivit y , we can assume that λ is a limit ordinal of uncountable cofinality . Set D α = U α \ S ξ<α U ξ for α < λ . Given p ositive µ ∈ C ( K ) ∗ , by the uncountable co finalit y , let α < λ such that µ ( D β ) = 0 fo r α ≤ β < λ . The regula rity of µ ensur es that µ ( D ) = 0, wher e D = S α ≤ β <λ D β . By inductive hypothesis, there e xist Bor el sets E , F suc h that E ⊆ S ξ<α H ξ ⊆ F and µ ( E ) = µ ( F ) = P ξ<α µ ( H ), so the co nclusion follows when we cons ide r E and F ∪ D .  It is evident that, g iven a legitimate system ( A n , H n ), the family H ′ n = H n ∪ { K \ A n , A n \ S H n } is scattered a nd has union K . Moreover, the family D n = { T i ≤ n H 1 ∩ . . . ∩ H n | H i ∈ H ′ i } also enjoys these prop erties. Readers familiar with rela ted literature will recog nis e that these families lea d directly to fra gmentabilit y , via Ribars k a ’s character is ation of fra gmentable spaces [1 0]. E le men ts o f the pro o f of the following r esult app ear in [15]. W e denote b oth canonica l norms o n C ( K ) and C ( K ) ∗∗ by || · || ∞ , and that of C ( K ) ∗ by || · || 1 . W e will b e iden tifying cer tain subsets of K with their indicator functions, either in C ( K ) or C ( K ) ∗∗ . Lemma 2.5. L et ( A n , H n ) b e a le gitimate system that sep ar ates p oints, with u nion H and which s atisfies pr op erties (1) – (3) ab ove. Then N = span ||·|| ∞ ( H ) is a sub algebr a of C ( K ) ∗∗ that is 1-norming for C ( K ) ∗ . Pr o of. Let D n be the families intro duced ab ov e, with union D . As H sepa - rates points, so do es D . If µ ∈ C ( K ) ∗ has v a riation | µ | then w e hav e || µ || 1 = GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 5 P D ∈ D n | µ | ( D ) by Lemma 2.4. Th us, g iven ε > 0 , we can take finite subs ets F n ⊆ D n and c ompact subsets K D ⊆ D , D ∈ F n , such that P n | µ | ( K \ S D ∈ F n K D ) < ε . Put M = T n S D ∈ F n K D and M D = M ∩ K D = M ∩ D . If M n = { M D | D ∈ F n } then M n is family of pairwis e disjoint sets with union M , and M n +1 refines M n . Moreover, each M D is clop en in M and, as D separ ates points of K , so M = S n M n separates p oints of M . Ther efore, b y the Stone-W eierstr ass Theo- rem, C ( M ) = span ||·|| ∞ ( M ). It follows that we ca n take no n-empty , disjoint M D i ∈ M and a i ∈ [1 , − 1], i ≤ n , such that | µ | ( M ) − P i ≤ n a i µ ( M D i ) < ε . No w M D i , M D j 6 = ∅ and M D i ∩ M D j = ∅ implies D i ∩ D j = ∅ . Ther efore, P i ≤ n | µ | ( D i \ M D i ) ≤ | µ | ( K \ M ) < ε . W e conclude that || µ || 1 − P i ≤ n a i µ ( D i ) < 2 ε . Since D ⊆ H , we are done.  W e say that a no r m || · || on X is p ointwise u niformly r otu nd , or p-UR , if there exists a se pa rating subspace F ⊆ X ∗ such that, g iven sequences ( x n ) and ( y n ) satisfying || x n || = || y n || = 1 and || x n + y n || → 2, then f ( x n − y n ) → 0 for all f ∈ F ; see, for example [1 2]. Evide ntly , p-UR norms ar e strictly conv ex . W e can now present the main theo rem. Theorem 2. 6. If K is a Gruenhage c omp act then: (1) C ( K ) ∗ admits an e quivalent, st rictly c onvex, dual lattic e norm; (2) C ( K ) ∗ admits an e quivalent, dual p-UR n orm. Pr o of. The la ttice norm is co nstructed first. W e take a legitima te system ( A n , H n ) satisfying the conclusion of Lemma 2.5. F or µ ∈ C ( K ) ∗ and m ≥ 1, define the seminorm || µ || 2 n,m = inf { m − 1 P H ∈ H n | λ | ( H ) 2 + || µ − λ || 2 1 | λ ∈ C ( A n ) ∗ } . W e observe that || µ || n,m ≤ || µ || 1 and that || · || n,m is w ∗ -low er semicontin uous. W e can verify the low er semicontin uity by applying a compac tness ar gument. Al- ternatively , if we denote the open set ( S H n ) ∪ ( K \ A n ) by U , we obs erve that || µ || n,m = sup { µ ( f ) | f ∈ B } , where B = { f ∈ C 0 ( U ) | m P H ∈ H n || f ↾ H || 2 ∞ + || f || 2 ∞ ≤ 1 } . In this way , we s ee that || · || n,m is a lso a la ttice seminorm. W e define a dual lattice norm on C ( K ) ∗ by se tting || µ || 2 = || µ || 2 1 + X n,m 2 − n − m || µ || 2 n,m . Now supp ose that || µ || = || ν || = 1 2 || µ + ν || . A standard conv exity a r gument (cf. [1, F act II.2 .3]) yields (1) 2 || µ || 2 n,m + 2 || ν || 2 n,m − || µ + ν || 2 n,m = 0 for all n and m . By app ealing to compactness or the Hahn-Banach Theo rem, ther e exist µ n,m , ν n,m ∈ C ( A n ) ∗ such that || µ || 2 n,m = m − 1 P H ∈ H n | µ n,m | ( H ) 2 + || µ − µ n,m || 2 1 and likewise for ν . Hence, by a pplying further sta ndard convexit y a rguments to equation (1), we obtain (2) 2 | µ n,m | ( H ) 2 + 2 | ν n,m | ( H ) 2 − | µ n,m + ν n,m | ( H ) 2 = 0 6 RICHARD J. SMITH for all n , m a nd H ∈ H n . Now we estimate || µ ↾ A n − µ n,m || 1 = || µ − µ n,m || 1 − || µ ↾ K \ A n || 1 ≤ || µ || n,m − || µ ↾ K \ A n || 1 ≤ [ m − 1 P H ∈ H n | µ | ( H ) 2 + || µ ↾ K \ A n || 2 1 ] 1 2 − || µ ↾ K \ A n || 1 ≤ m − 1 2 bec ause || · || 1 ≤ || · || . A similar result holds for ν . Therefore, we conclude from equation (2) that 2 | µ | ( H ) 2 + 2 | ν | ( H ) 2 − | µ + ν | ( H ) 2 = 0 for all H ∈ H n and n ∈ N . As N from Lemma 2.5 is no rming, we certainly obtain | µ | = | ν | = 1 2 | µ + ν | . This gives µ = ν by the following lattice argument, included for completeness. If λ = µ + − ν − then | µ | = | ν | implies λ = ν + − µ − , meaning µ + ν = 2 λ . Hence µ + + µ − = | µ | = 1 2 | µ + ν | = λ + + λ − . W e s ee that λ + = ( µ + − ν − ) + ≤ ( µ + ) + = µ + , hence µ + = λ + and µ − = λ − . W e co nc lude that µ = ν as cla imed. Now we co nstruct the p-UR norm, using the norming subspace N . Fir st, w e claim tha t || · || a b ove alr eady satisfies the p-UR pro per t y if µ k and ν k are p ositive. Suppo se that µ k and ν k are p ositive measures such that || µ k || = || ν k || = 1 and || µ k + ν k || → 2. As a b ove, we ca n find µ k,n,m , ν k,n,m ∈ C ( A n ) ∗ such that || µ || 2 k,n,m = m − 1 P H ∈ H n | µ k,n,m | ( H ) 2 + || µ k − µ k,n,m || 2 1 and likewise fo r ν k . B y co nvexit y a rguments, we o btain (3) 2 | µ k,n,m | ( H ) 2 + 2 | ν k,n,m | ( H ) 2 − | µ k,n,m + ν k,n,m | ( H ) 2 → 0 as k → ∞ . Moreov er , if H ∈ H n , we e s timate | µ k − µ k,n,m | ( H ) ≤ || µ k ↾ A n − µ k,n,m || 1 ≤ m − 1 2 and likewise for ν k . Therefor e, by fixing m large enough and app ealing to equatio n (3), we get 2 µ k ( H ) 2 + 2 ν k ( H ) 2 − ( µ k + ν k )( H ) 2 → 0 whence ( µ k − ν k )( H ) → 0. It follows that ξ ( µ k − ν k ) → 0 for all ξ ∈ N , thus completing the claim. Now we set ||| µ ||| 2 = || µ + || 2 + || µ − || 2 . T o see that this defines a dual no rm, observe that as || · || is a la ttice no rm, we have || µ + || = sup { µ ( f ) | f ∈ C ( K ) , f ≥ 0 and || f || ≤ 1 } where || · || also denotes the predual nor m. Thus µ 7→ || µ + || is w ∗ -low er semicon- tin uous, and likewise for µ 7→ || µ − || . Now, g iven g e ne r al µ k and ν k satisfying 2 ||| µ k ||| 2 + 2 ||| ν k ||| 2 − ||| µ k + ν k ||| 2 → 0 we g et 2 || ( µ k ) + || 2 + 2 || ( ν k ) + || 2 − || ( µ k ) + + ( ν k ) + || 2 → 0 and simila rly for ( µ k ) − and ( ν k ) − . Therefore, we c an apply the claim twice to get ( µ k − ν k )( H ) → 0 for all H ∈ H .  GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 7 W e a pply the theorem a bove to obtain renorming res ults for mo r e gener al Bana ch spaces. First, we give a mo dest generalis a tion of the classic transfer method for LUR reno r mings, applied to strictly conv ex reno rmings; cf. [1, Theor e m I I.2 .1 ]. A pro of is pr ovided for completeness. Prop ositio n 2. 7 . L et ( X , || · || ) ∗ , ( Y , || · || ) ∗ b e dual Banach sp ac es, with ( Y , || · || ) ∗ strictly c onvex . F urt her, let ||| · ||| b e a c o arser, w ∗ -lower semic ontinuous seminorm on X ∗ , T : X ∗ − → Y ∗ a b oun de d, line ar op er ator and set Z = T ∗ Y ∗ |||·||| ⊆ X ∗ . Then ther e exists an e quivalent dual norm | · | on X , such that whenever f ∈ Z, f ′ ∈ X and | f | = | f ′ | = 1 2 | f + f ′ | we have ||| f − f ′ ||| = 0 . Pr o of. Define semino r ms | · | n on X ∗ by | f | 2 n = inf {||| f − T ∗ g ||| + n − 1 || g || | g ∈ Y ∗ } and set | f | 2 = || f || 2 + P n ≥ 1 2 − n | f | 2 n . Since ||| · ||| is coar s er than || · || , our new nor m | · | is equiv alent to || · || . As in Theorem 2.6, by a w ∗ -compactness argument o r the Hahn-Ba nach Theorem, | · | n is a w ∗ -low er semicontin uo us seminorm, and the infim um in the de finitio n is atta ined. Now let f and f ′ satisfy the ab ov e hypothesis. By convexit y arg umen ts and infimum attainment, we can take g n , g ′ n ∈ Y ∗ such that (4) | f | 2 n = ||| f − T ∗ g n ||| 2 + n − 1 || g n || 2 , (5) ||| f − T ∗ g n ||| = ||| f ′ − T ∗ g ′ n ||| and || g n || = || g ′ n || = 1 2 || g n + g ′ n || . The las t equation tells us that g n = g ′ n for all n , meaning that we hav e ||| f − f ′ ||| ≤ ||| f − T ∗ g n ||| + ||| f ′ − T ∗ g ′ n ||| Since f ∈ Z , we have | f | n → 0 , so by equations (4) and (5), this leads to ||| f ′ − T ∗ g ′ n ||| = ||| f − T ∗ g n ||| → 0, g iving ||| f − f ′ ||| = 0 as r e q uired.  Using this, we can obtain our genera l renorming result. Prop ositio n 2.8 . L et ( X , || · || ) b e a Banach sp ac e, F ⊆ X ∗ a subsp ac e and ||| · ||| a c o arser norm on X , su ch that F ∩ ( X , ||| · ||| ) ∗ sep ar ates p oints of X . F urther, let K ⊆ X b e a Gruenhage c omp act in the σ ( X , F ) -top olo gy and su pp ose X = span |||·||| ( K ) . Then: (1) ther e is a c o arser, σ ( X , F ) -lower semic ontinuous, st rictly c onvex norm | · | on X ; (2) X admits an e quivalent, st rictly c onvex norm. Mor e over, if F is a norming subsp ac e t hen | · | is e qu ivalent to || · || . Pr o of. Since F is separating, we can identify (( X , || · || ) , σ ( X, F )) as a top olog ical subspace o f (( F , || · || ) ∗ , w ∗ ) by standa rd ev aluatio n and consider K a s a w ∗ -compact subset of F ∗ . Now elements of F act as c o nt inuous functions on ( K , w ∗ ) and the map S : C ( K ) ∗ − → F ∗ , given by ( S µ )( f ) = R K f d µ , is a dual op erator. Let ||| · ||| also denote the canonical no rm on G = ( X , ||| · ||| ) ∗ , and define the w ∗ -low er semicontin uous seminorm ||| ξ ||| 1 = s up { ξ ( f ) | f ∈ F a nd ||| f ||| ≤ 1 } 8 RICHARD J. SMITH on F ∗ . B y Pr op osition 2.7 , there exists an e quiv alent, dual norm | · | 1 on F ∗ , such that if ξ ∈ S C ( K ) ∗ |||·||| 1 , ξ ′ ∈ F ∗ and | ξ | 1 = | ξ ′ | 1 = 1 2 | ξ + ξ ′ | 1 then ||| ξ − ξ ′ ||| 1 = 0. Let | · | b e the restrictio n of | · | 1 to X a nd note that | · | is b oth σ ( X , F )-lower semicontin uous and coarser than ||· || . Mor eov er , X = span |||·||| ( K ) ⊆ S C ( K ) ∗ |||·||| 1 . Therefore, whenever | x | = | x ′ | = 1 2 | x + x ′ | , w e hav e ||| x − x ′ ||| 1 = 0. Since F ∩ G separates p oints o f X , it fo llows that x ′ = x . This gives (1). F or (2), o bs erve that the sum || · || + | · | is a n equiv alent, s trictly conv ex norm o n X . Finally , if F is norming then | · | is e quiv alent to || · || .  Let us as sume that the coa r ser nor m ||| · ||| o f Pro po sition 2.8 is σ ( X, F )- lower semicontin uous. B y a standard p ola r a rgument ||| x ||| = sup { f ( x ) | f ∈ F , ||| f ||| ≤ 1 } and, in pa rticular, F ∩ ( X, ||| · ||| ) ∗ separates p oints o f X . Corollary 2.9. L et X b e a Banach sp ac e and X ∗ = span |||·||| ( K ) , wher e K is a Gruenhage c omp act in the w ∗ -top olo gy and ||| · ||| is e quivalent to a c o arser, w ∗ - lower semic ontinuous norm on X ∗ . Then X ∗ admits an e quivalent, s t rictly c onvex dual n orm. The result ab ov e applies to all esta blis hed clas ses of Ba nach spaces known to admit equiv a lent str ictly co n vex dua l nor ms on their dual spaces; for example, V a ˇ s´ ak spaces. W e move o n to discuss a prop erty o f B a nach spaces , in tro duced in [3] a nd shown there to b e a sufficient condition for the existence of an equiv a len t, strictly conv ex dual norm. Definition 2. 1 0 ([3 ]) . W e say tha t the Banach spa c e X has pro per ty G if there exists a b ounded set Γ = S n ∈ N Γ n ⊆ X , with the prop erty that whenever f , g ∈ B X ∗ are distinct, there exist n ∈ N and γ ∈ Γ n such that ( f − g )( γ ) 6 = 0 and, either | f ( γ ′ ) | > 1 4 | ( f − g )( γ ) | for finitely ma ny γ ′ ∈ Γ n , or | g ( γ ′ ) | > 1 4 | ( f − g )( γ ) | for finitely many γ ′ ∈ Γ n . As well as showing that all V a ˇ s´ ak spaces p o ssess pr o pe rty G, the a uthors of [3] remark that the prop erty is closely related to Gruenhag e compacta. Prop ositio n 2.11. If X has pr op erty G t hen the dual u nit b al l B X ∗ is a Gruenhage c omp act in the w ∗ -top olo gy. Pr o of. W e ca n and do a ssume that Γ is a subse t o f the unit ball B X . Given γ ∈ Γ and q ∈ (0 , 1) ∩ Q , we let U ( γ , q ) = { f ∈ B X ∗ | f ( γ ) > q } . W e pr ov e that, together , ( U n,q ) and ( V n,q ), n ∈ N and q ∈ (0 , 1) ∩ Q , sa tisfy (1) and (2) of Definition 1 .1, where U n,q = { U ( γ , q ) | γ ∈ Γ n } and V n,q = { − U ( γ , q ) | γ ∈ Γ n } . Given distinct f , g ∈ B X ∗ , take γ ∈ Γ n with the prop erty that α = 1 4 | ( f − g )( γ ) | > 0. It follows that either | f ( γ ) | > α or | g ( γ ) | > α ; without lo ss of generality , we assume that the former inequa lit y ho lds . No w supp os e that f ( γ ) > 0. W e choos e rational q to satisfy f ( γ ) > q > max { g ( γ ) , α } if f ( γ ) > g ( γ ), or g ( γ ) > q > f ( γ ) otherwise. Either wa y , U ( γ , q ) ∩ { f , g } is a singleton, giving (1). Since q > α , (2) fo llows. If f ( γ ) < 0, we rep eat the ab ov e argument w ith − f a nd − g .  GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 9 Corollary 2.12 ([3]) . If X ha s pr op ert y G t hen X ∗ admits an e quivalent, st rictly c onvex dual norm. Pr o of. Combine Pr op osition 2.1 1 and Cor ollary 2.9.  W e finish this section with an o pen pro ble m. Problem 2.13. If C ( K ) ∗ admits a strictly c onvex dual norm then is K Gruenhage? Mor e ambitiously, if X ∗ is a dual Banach sp ac e with strictly c onvex dual norm, is B X ∗ Gruenhage? 3. A to pological characterisa tion of Y -embedd able trees In this s ection, we pr esent a pa rtial conv er se to The o rem 2.6. W e call a pa rtially ordered se t (Υ , 4 ) a tr e e if, for each t ∈ Υ, the set (0 , t ] = { s ∈ Υ | s 4 t } of pr e de c essors o f t is well-ordered. Given t ∈ Υ, we denote by t + the set of imme diate suc c essors of t in Υ; that is, u ∈ t + if and only if t ≺ u and t ≺ ξ ≺ u for no ξ . The lo cally compact, scattered or der top olo gy on Υ takes a s a ba sis the sets ( s, t ], s ≺ t , where ( s, t ] = (0 , t ] \ (0 , s ]. T o ensure that this topo lo gy is als o Hausdorff, we dema nd that every no n-empty , totally order ed subs e t of Υ has at most one minima l upp er bo und; trees satisfying this pro per t y ar e themselves called Hausdorff . W e study the space C 0 (Υ) of contin uous, real-v a lued functions on Υ that v a nish at infinit y , and the dual spa c e o f mea s ures. T o date, most of the results ab out renorming C 0 (Υ) and its dual hav e b een order- theoretic in character: [5], [13] a nd [14]. Such order-theo retic results, while well-suited in this context, are deeply b ound to the tree-structure and, a s s uch, do not offer obvious generalis ations. Here, we are able to give a purely top o logical characterisation of tree s Υ, such that C 0 (Υ) ∗ admits an equiv a lent , strictly conv ex dual nor m. The following definition first app ears in [13]. Definition 3.1. Let Y b e the set of all strictly inc r easing, contin uo us, transfinite sequences x = ( x α ) α ≤ β of r e al num b ers, where 0 ≤ β < ω 1 . W e o rder Y by declaring that x < y if and only if either y strictly extends x , or if there is so me ordinal α s uch that x ξ = y ξ for ξ < α and y α < x α . W e say that a ma p ρ : Υ − → Σ from a tree to a linear order is incr e asing if ρ ( s ) ≤ ρ ( t ) whenever s ≺ t , and strictly s o if the fo rmer inequality is always strict. The next theo rem is the key result of this section. Theorem 3.2. If Υ is a tre e and ρ : Υ − → Y is a strictly incr e asing function then Υ is a Gruenhage sp ac e. Theorem 2.6 and T he o rem 3.2, tog ether with [1 3, Pr op osition 7 ] and a res ult from [14], a llows us to present the following serie s of equiv alent c o nditions a nd, in particular, provides our partial conv er se to The o rem 2.6. Obser ve that a lo cally compact space is Gr uenhage if and o nly if its 1 -p oint co mpactification is. Corollary 3.3. If Υ is a tr e e then the fol lowing ar e e quivalent: (1) C 0 (Υ) ∗ admits an e quivalent, dual p-UR norm; (2) C 0 (Υ) ∗ admits an e quivalent, strictly c onvex dual lattic e n orm; (3) C 0 (Υ) admits an e quivalent, Gˆ ate aux smo oth lattic e norm; (4) C 0 (Υ) ∗ admits an e quivalent, strictly c onvex dual norm; (5) ther e is a strictly incr e asing fun ction ρ : Υ − → Y ; 10 RICHARD J. SMITH (6) Υ is a Gruenhage sp ac e. It is prov ed in [1 3] that the 1-p oint c o mpactification of a tree Υ is descrip- tive, equiv ale ntly σ -discrete, if and only if there is a strictly increas ing function ρ : Υ − → Q . As tree s g o, thos e that admit such Q -v alued functions are rela- tively simple. The order Y is considera bly larger than Q in order -theoretic terms; indeed, given any o rdinal β < ω 1 , the lex icographic pro duct R β embeds into Y . Accordingly , there is an a bundance of trees tha t admit str ictly incr e a sing Y -v alued maps, but no t strictly inc r easing Q -v a lued maps [13]. Therefore, the class of Gru- enhage compa c t spaces encompasses a ppreciably mor e structure than the class of descriptive co mpact spa c es. A little prepa ratory work must b e presented b efore giving the pr o of of Theorem 3.2. W e r ecall so me materia l fro m [13]. Definition 3.4 ([13]) . A s ubs et V ⊆ Υ is called a plate au if V ha s a least element 0 V and V = S t ∈ V [0 V , t ]. A pa rtition P o f Υ consisting solely of plateaux is called a plate au p artition . If V is a plateau then V \{ 0 V } is o p en, so given a plateau par tition P of Υ, the set H = { 0 V | V ∈ P } of le ast elements of V is close d in Υ. Definition 3.5 ([13]) . Given a tree Υ, let ( P β ) β <ω 1 be a seq ue nc e of plateau partitions with the following prop erties: (1) if α < β and V ∈ P α , W ∈ P β , then either W ⊆ V or V ∩ W is empt y; (2) if β is a limit o rdinal and W ∈ P β , then W = \ { V | V ∈ P α , α < β , W ⊆ V } ; (3) if t ∈ Υ , there e x ists β < ω 1 , dep ending o n t , such that { t } ∈ P β . W e call such a s equence o f plateau partitions admissible . Definition 3. 6 ([13 ]) . Let ( P β ) β <ω 1 be admissible and let T b e the tree { ( α, V ) | V ∈ P α , α < ω 1 } with order ( α, V ) ≺ ( β , W ) if and only if α ≤ β and W ⊆ V . Then the s ubtree Υ( P ) = { ( β , V ) ∈ T | U is no t a s ingleton whenever ( α, U ) ≺ ( β , V ) } of T is ca lled the p artition tr e e of Υ with resp ect to ( P β ) β <ω 1 . It is ev iden t that if V is a plateau then s o is V , with 0 V = 0 V . A subset of a tree Υ is called a n antichain if it consists s olely of pairwis e incompara ble elements. With resp ect to the interv al top ology , a ntic hains are discrete subs ets. W e make the following elementary , yet impo rtant, obser v ation. Lemma 3 .7. Le t E b e an antichain in a p artition tr e e Υ( P ) . If ( α, V ) and ( β , W ) ar e distinct elements of E then b oth interse ct ions V ∩ W and V \{ 0 V } ∩ W \{ 0 W } ar e empty. Pr o of. W e can assume that α ≤ β . Tha t the first intersection is empty follows directly from the definition o f the partition tree or der. T o see that the same is true for the second, note that if ( α, U ) 4 ( β , W ) then W \{ 0 W } ⊆ U \{ 0 U } , so all we need to do is prov e that if t ∈ V \{ 0 V } ∩ U \ { 0 U } then V and U intersect non- trivially and a re th us equal. Given such t , we hav e that 0 V and 0 U are compar a ble. If 0 V 4 0 U then since there exists s ∈ (0 U , t ] ∩ V , we hav e 0 U ∈ V a s V is a plateau. Likewise, if 0 U 4 0 V then 0 V ∈ U .  GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 11 The next result shows that if there is a s trictly incr easing function ρ : Υ − → Y then Υ admits a partition tre e Υ ( P ), on which may b e defined a strictly incr easing, real-v alued function. It is imp ortant to note that the or der o f the partition tree is related to the order o f Υ thro ugh the seco nd, alb eit technical, pr op erty b elow. If t ∈ Υ then the we dge [ t, ∞ ) is the set { u ∈ Υ | u < t } . Prop ositio n 3.8 ([13]) . L et Υ b e a tr e e. If ρ : Υ − → Y is strictly incr e asing then ther e ex ists an admissible se quenc e of p artitions ( P β ) β <ω 1 that yields a p artition tr e e Υ( P ) , and a st rictly incr e asing function π : Υ( P ) − → [0 , 1] . Mor e over: (1) P 0 = { [ r, ∞ ) | r ∈ Υ is minimal } ; (2) for any non-maximal ( β , V ) ∈ Υ( P ) , t he map 0 W 7− → π ( β + 1 , W ) is s t rictly de cr e asing on t he su btr e e of le ast elements H ( β ,V ) = { 0 W | ( β + 1 , W ) ∈ ( β , V ) + } . In the pr o of b elow, we will assume the par tition tree Υ( P ) and function π from Prop ositio n 3.8. Pr o of of The or em 3.2. W e construct a leg itimate sy s tem on Υ . As Υ( P ) admits a strictly incr e asing, real-v a lued function π , its isolated elements may b e dec o mpo sed int o a co unt able union o f antic hains ( F n ). Indeed, if ( β , W ) ∈ Υ( P ) is iso lated and no n-minimal, then it has an immediate predecess or ( α, V ), and we can pick τ ( β , W ) ∈ Q ∩ ( π ( α, V ) , π ( β , W )). Then consider the a nt ichain of minimal elements, together with the fibres ( τ − 1 ( q )) q ∈ Q . If V is a plateau then V \ V is a n antic hain and hence discr ete. Note that here, clos ure is taken with resp ect to Υ. F rom Lemma 3.7, the family { V \{ 0 V } | ( β , V ) ∈ F n } is a pa irwise disjoint collection o f op en sets in Υ. Hence D n = S { V \ V | ( β , V ) ∈ F n } is discrete. Given q ∈ Q , co nsider the s et E q of s uc c e ssor e lemen ts ( β + 1 , W ) ∈ ( β , V ) + , with ( β , V ) ∈ Υ( P ) ar bitrary , such that π ( β , V ) < q < π ( β + 1 , W ). O bserve that E q is an antic ha in in Υ ( P ). Indeed, if ( α + 1 , U ) ≺ ( β + 1 , W ) and ( β + 1 , W ) ∈ E q then ( α + 1 , U ) 4 ( β , V ) ≺ ( β + 1 , W ), thu s π ( α + 1 , U ) ≤ π ( β , V ) < q . It follows that ( α + 1 , U ) / ∈ E q . Given non- maximal ( β , V ) ∈ Υ( P ), prop erty (2) of Prop ositio n 3.8 tells us that, in pa rticular, the set of rela tively iso la ted p oints in the lea st elements H ( β ,V ) can be deco mpo s ed into a co unt able union o f antic hains ( F ( β ,V ) ,m ) in Υ. Given ( β + 1 , W ) ∈ ( β , V ) + such that 0 W ∈ F ( β ,V ) ,m , set E q, ( β ,V ) , W = { ( β + 1 , W ′ ) ∈ E q ∩ ( β , V ) + | 0 W 4 0 W ′ } and E q,m = { E q, ( β ,V ) , W | ( β + 1 , W ) ∈ ( β , V ) + and 0 W ∈ F ( β ,V ) ,m } . W e observe that ea ch E q,m is a family of disjoint subse ts of E q . Indeed, let E q, ( β ,V ) , W , E q, ( β ′ ,V ′ ) ,W ′ ∈ E q,m . If ( β , V ) 6 = ( β ′ , V ′ ) then ( β , V ) + ∩ ( β ′ , V ′ ) + is empt y and we a re done, so we a s sume that this is not the c a se. If W 6 = W ′ then 0 W and 0 W ′ are incomparable in Υ, so E q, ( β ,V ) , W and E q, ( β ′ ,V ′ ) ,W ′ m ust be disjo in t. By Le mma 3.7, it follows that the sets J q, ( β ,V ) , W = [ { W ′ | ( β + 1 , W ′ ) ∈ E q, ( β ,V ) , W } , E q, ( β ,V ) , W ∈ E q,m , ar e a ls o pairwise disjoint. W e prove that J = J q, ( β ,V ) , W is a plateau. E vidently 0 W is the leas t ele ment of J . No w supp ose t ∈ J and 0 W 4 s 4 t . W e have to sho w that s ∈ J . As 12 RICHARD J. SMITH 0 W , t ∈ V and V is a plateau, s ∈ V and so there ex ists ( β + 1 , W ′ ) ∈ ( β , V ) + such that s ∈ W ′ . W e know that t ∈ W ′′ , where ( β + 1 , W ′′ ) ∈ E q ∩ ( β , V ) + and 0 W 4 0 ′′ W . Th us we hav e 0 W 4 0 W ′ 4 0 W ′′ and, b y co ndition (2) of Prop osition 3.8, π ( β + 1 , W ′ ) ≥ π ( β + 1 , W ′′ ) > q . It follows that ( β + 1 , W ′ ) ∈ E q and s ∈ J . A t last, we have eno ugh informa tion to define our legitimate s ystem. B e gin by setting A = Υ a nd H = {{ t } | t ∈ Υ is iso lated } . Then define A n = D n and H n = {{ t } | t ∈ D n } . Aga in using Lemma 3.7, we are p ermitted to define A ′ n = Υ and H ′ n = { V \{ 0 V } | ( β , V ) ∈ F n } . F rom the ab ove disc ussion, given q ∈ Q and m ∈ N , we can define A q,m = Υ and H q,m = { J q, ( β ,V ) , W \{ 0 W } | E q, ( β ,V ) , W ∈ E q,m } . W e claim that, together, the families H , H n , H ′ n and H q,m separate p oints of Υ in the manner of P rop osition 2.1, part (2). Let s, t be distinct elemen ts of Υ . If s or t is an isolated p oint of Υ, we can se pa rate using H . Henceforth, we will assume that b oth s and t are limit elements of Υ . L et V s β be the unique element o f β containing s , and likewise for t . Let γ < ω 1 be minimal, s ub ject to the co ndition that V s γ 6 = V t γ . Such γ exists by prop erty (3) of Definition 3.5. By prop erty (2) of Definition 3.5, γ ca nno t b e a limit or dinal. If γ = 0 then V = V s γ = [ r, ∞ ) by prop erty (1) o f Prop ositio n 3.8. Being minimal in Υ, r is isola ted, so s ∈ V \{ 0 V } . As (0 , V ) is minimal in Υ ( P ), it is an element o f F n for some n . Consequent ly , we can s eparate s from t using H ′ n . W e finish by tackling the c a se where γ = β + 1 for some or dinal β . Le t W = V s β +1 and W ′ = V t β +1 . If s ∈ W \ { 0 W } then as ( β + 1 , W ) is isolated in Υ( P ), we can separ ate using some H ′ n as ab ove. W e can argue similarly if t ∈ W ′ \{ 0 W ′ } so, from now on, we assume that s = 0 W and t = 0 W ′ , i.e. s, t ∈ H ( β ,V ) . If 0 W is a n immediate success or with resp ect to H ( β ,V ) , i.e. if there exists 0 U ∈ H ( β ,V ) such that 0 U ≺ 0 W and no element of H ( β ,V ) lies str ictly b et ween the t wo, then 0 W ∈ U \ U . Indee d, if r ≺ 0 W then as 0 W is a limit in Υ, there exists ξ ∈ (max { r , 0 U } , 0 W ] \{ 0 W } . Now ξ must lie in U because 0 U is the immediate predecessor of 0 W in H ( β ,V ) . It follows that 0 W ∈ U as requir e d. Now ( β + 1 , U ) is in F n for so me n , so { 0 W } ∈ H n , thus sepa rating 0 W from 0 W ′ . As abov e, we can ar gue similarly if 0 W ′ is an immediate successor with res pect to H ( β ,V ) , so now we assume tha t neither 0 W nor 0 W ′ are such elements. As H ( β ,V ) has a least element a nd is a Hausdorff tree in its own right, the grea test element less than bo th 0 W and 0 W ′ is some 0 U ∈ H ( β ,V ) and, without los s of generality , we can assume tha t 0 U ≺ 0 W . If 0 U ′ is the immediate s uccessor of 0 U in H ( β ,V ) then 0 U ′ ≺ 0 W , b ecause 0 W is not such an element. Cons e quent ly , 0 U ′ ∈ F ( β ,V ) ,m for some m so, given r ational q strictly b etw een π ( β , V ) a nd π ( β + 1 , W ), w e ha ve 0 W ∈ J \ { 0 U ′ } , where J = J q, ( β ,V ) , U ′ . Since 0 U ′ 6 4 0 W ′ by maximality o f 0 U , it follows that J \{ 0 U ′ } s eparates 0 W from 0 W ′ .  4. St abil ity proper ties of Gruenha ge sp aces Our first stability prop er ty is purely top olog ical. Theorem 4 . 1. If X is a Gruenhage sp ac e and f : X − → Y is a p erfe ct, surje ctive mapping, then Y is also Gruenhage. Pr o of. Let X be a Gruenha g e space and as sume that we hav e families ( U n ) and sets R n satisfying Pro po sition 2.1 (3). By adding new families { S U n } if necess a ry , GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 13 we a s sume that given n , there exists m such that R m = S U n . If G ⊆ N is finite, define V G = { T i ∈ G U i | ( U i ) i ∈ G ∈ Q i ∈ G U i } . Given a p erfect, sur jective map f : X − → Y , we set V F, G,k = { Y \ f ( X \ ( S i ∈ F R i ∪ S F )) | F ⊆ V G and ca rd F = k } for finite F, G ⊆ N a nd k ∈ N . Since f is per fect, every element o f V F, G,k is op en in Y . Let y , z ∈ Y b e distinct. W e show that there exists finite F , G ⊆ N , k ∈ N and F ⊆ V G with cardinality k , such that { y , z } ∩ Y \ f ( X \ ( S i ∈ F R i ∪ S F )) is a sing leton. Mo reov er, if G ⊆ V G has cardinality k and y ∈ Y \ f ( X \ ( S i ∈ F R i ∪ S F )) is no n-empty , then G = F . F rom this, it follows immediately that Y is Gr uenhage. T o prov e this cla im, we first constr uct a pair of decrea sing sequences o f compa ct sets. Set A 0 = f − 1 ( y ) and B 0 = f − 1 ( z ). Given r ≥ 0 , if, for a ll n , it is true that ( A r ∪ B r ) ∩ R n = ∅ or A r ∩ B r ⊆ R n , then we stop. If not then let n r +1 be minimal, sub ject to the requir ement that ( A r ∪ B r ) ∩ R n r +1 6 = ∅ and ( A r ∪ B r ) \ R n r +1 6 = ∅ . Put A r +1 = A r \ R n r +1 and B r \ R n r +1 . Contin uing in this wa y , either we stop at a finite stage o r co nt inue indefinitely . If the pr o cess stops at a finite stage r ≥ 0, set A = A r and B = B r . Evidently ( A ∪ B ) ∩ R n = ∅ or A ∪ B ⊆ R n for all n . If the pro cess ab ov e con tinues indefinitely , then we obtain a sequence n 1 < n 2 < . . . and decreasing sequences ( A i ), ( B i ) of non-empty , compa ct sets. Put A = T ∞ i =0 A i and B = T ∞ i =0 B i . Then, given any n , again we have ( A ∪ B ) ∩ R n = ∅ o r A ∪ B ⊆ R n , lest we vio late the minimality of the first n i > n . If A = ∅ then by s urjectivity , and compactness if necessa r y , there is some r ≥ 1 such that A r = ∅ . Since ( A r − 1 ∪ B r − 1 ) \ R n r 6 = ∅ by construction, it is not the case that B r is empty , thus f − 1 ( y ) ⊆ S r i =1 R n i and f − 1 ( z ) 6⊆ S r i =1 R n i . P ut F = { n 1 , . . . , n r } a nd let G b e arbitr ary . Then Y \ f ( X \ S i ∈ F R i ) is the o nly element of V F, G, 0 and { y , z } ∩ Y \ f ( X \ S i ∈ F R i ) = { y } . If B = ∅ then we pro ceed similar ly . Now s upp os e that A 6 = ∅ a nd B 6 = ∅ . Define K = A ∪ B and let I = { n ∈ N | K ∩ R n = ∅ a nd K ⊆ S U n } . W e hav e K = S { K ∩ U | U ∈ U n } whenever n ∈ I . Moreover, the sets in ea ch { K ∩ U | U ∈ U n } , n ∈ I , ar e pairwise disjo in t. In fact slightly mor e can be sa id; if x ∈ K ∩ U ∩ V for U, V ∈ U n and n ∈ I then U = V . Indeed, if x ∈ K ∩ U ∩ V , U, V ∈ U n and U 6 = V then x ∈ R n , so n 6∈ I . Given distinct a, b ∈ K , there exis ts n and U ∈ U n such that { a , b } ∩ U is a singleton. Firstly , this means n ∈ I . Indeed, if K ∩ R n 6 = ∅ then K ⊆ R n , meaning a, b ∈ U , which is not the case. Now supp ose K 6⊆ S U n . W e hav e S U n = R m for s ome m , so K ∩ S U n = K ∩ R m is empty , w hich again is not the case. Thus n ∈ I . In particular, this means we can ass ume that { a, b } ∩ U = { a } b ecause 14 RICHARD J. SMITH { K ∩ V | V ∈ U n } partitions K . By compactness , it follows that there is finite G ⊆ I and finite E ⊆ S i ∈ G U i such that A ⊆ [ { K ∩ U | U ∈ E } and B 6⊆ [ { K ∩ U | U ∈ E } . Let x ∈ K ∩ U , whe r e U ∈ E . F or every i ∈ G , we know fr om ab ove that there is a unique U i ∈ U i such that x ∈ U i . By definition T i ∈ G U i ∈ V G , and s ince U ∈ U j for some j ∈ G , we have U = U j and x ∈ T i ∈ G U i ⊆ U . This allows us to take a finite subset F ⊆ V G , such tha t A ⊆ [ { K ∩ V | V ∈ F } and B 6⊆ [ { K ∩ V | V ∈ F } . W e choose F so that it has minimal car dinality k . If necessary , we a ppe a l to c ompactness to find r ≥ 0 s atisfying f − 1 ( y ) ⊆ r [ i =1 R n i ∪ [ F , A ⊆ f − 1 ( y ) \ S r i =1 R n i and B ⊆ f − 1 ( z ) \ S r i =1 R n i . Let F = { n 1 , . . . , n r } . Observe that if G ⊆ V G and f − 1 ( y ) ⊆ S i ∈ F R i ∪ S G then A ⊆ S G , and likewise for f − 1 ( z ) and B . Th us f − 1 ( z ) 6⊆ S i ∈ F R i ∪ S F and consequently { y , z } ∩ Y \ f ( X \ ( S i ∈ F R i ∪ S F )) = { y } . Now let y ∈ Y \ f ( X \ ( S i ∈ F R i ∪ S G )), wher e G ⊆ V G has car dinality k . It fo llows that A ⊆ S G . W e show that G = F . T ak e W ∈ F . By minimality of k A 6⊆ [ { K ∩ V | V ∈ F \ { W }} th us ther e is x ∈ A ∩ W . T a ke V ∈ G such that x ∈ V . W e claim tha t W = V . Indeed, W = T i ∈ G W i and V = T i ∈ G V i for some W i , V i ∈ U i , i ∈ G . Since G ⊆ I and x ∈ K ∩ W i ∩ V i , we hav e W i = V i for all i ∈ G , hence W = V ∈ G . Therefore F ⊆ G and, by cardinality , we have eq ua lit y as requir ed.  Next, something of a more functional ana lytic natur e . Prop ositio n 4 .2. If K is a Gruenhage c omp act t hen so is B C ( K ) ∗ . Pr o of. Let ( A n , H n ) b e a legitimate sys tem satisfying prop erties (1) – (3), presented after C o rollar y 2 .3. W e can and do assume that ∅ ∈ H n for all n . Giv en H ∈ H n and q ∈ (0 , 1 ) ∩ Q , define the w ∗ -op en set U ( H,n,q ) + = { µ ∈ B C ( K ) ∗ | µ + ( H ∪ ( K \ A n )) > q } and let U ( n,q ) + = { U ( H,n,q ) + | H ∈ H n } . Define U ( H,n,q ) − and U ( n,q ) − in the cor re- sp onding manner . W e claim that, with resp ect to U ( n,q ) + and U ( n,q ) − , n ∈ N and q ∈ (0 , 1) ∩ Q , B C ( K ) ∗ is a Gruenhage compact in the sense of Definition 1.1. Let µ, ν ∈ B C ( K ) ∗ be distinct. Either µ + 6 = ν + or µ − 6 = ν − . W e supp ose that the former holds; if the latter holds then we rep eat the arg umen t b elow using the sets U ( H,n,q ) − and U ( n,q ) − . By Lemma 2.5 , ther e exists n ∈ N and H 0 ∈ H n such that µ + ( H 0 ) 6 = ν + ( H 0 ). If µ + ( K \ A n ) 6 = ν + ( K \ A n ) then set H = ∅ . Otherwise, set H = H 0 . Either wa y , we hav e µ + ( H ∪ ( K \ A n )) 6 = ν + ( H ∪ ( K \ A n )) a nd, witho ut loss of ge ne r ality , we supp o se that µ + ( H ∪ ( K \ A n )) < q < ν + ( H ∪ ( K \ A n )) for some ra tional q . Then { µ, ν } ∩ U ( H,n,q ) + = { ν } . Moreover, if µ ∈ U ( H ′ ,n,q ) + for some H ′ ∈ H n then µ + ( H ′ ) = µ + ( H ′ ∪ ( K \ A n )) − µ + ( K \ A n ) > q − µ + ( H ∪ ( K \ A n )) > 0. GR UENHAGE COMP A CT A AND STRICTL Y CONVEX DUAL NORMS 15 Hence, as ea ch H n is a fa mily of pairwise disjo int sets, µ can only be in finitely many e le men ts o f U ( n,q ) + .  W e finish b y using these tw o results to g lean a further cr op of stability pro p er ties. Prop ositio n 4 .3. (1) If K is a Gruenhage c omp act and π : K − → M is c ontinu ous and surje ctive then M is also Gruenhage; (2) if X n , n ∈ N ar e Gruenhage sp ac es then so is Q n X n ; (3) if X is a Banach sp ac e, F ⊆ X ∗ is a sep ar ating subsp ac e and K ⊆ X is a Gruenhage c omp act in the σ ( X , F ) -top olo gy t hen s o is its symmetric, σ ( X , F ) -close d c onvex hul l. Pr o of. (1) follows immediately from Theorem 4.1. T o pr ove (2), we let X n hav e a sequence ( U n,m ) m ∈ N of families of op en sets s atisfying Definition 1.1. It is str aight- forward to verify tha t the families ( V n,m ), defined by V n,m = { Q in X i | U ∈ U n,m } are witnes s to the fact that Q n ∈ N X n is Gr uenhage. T o see that (3) ho lds, c o nsider, as in P rop osition 2.8, K as a subset of F ∗ and the map S res tricted to B C ( K ) ∗ , which is Gruenhage by Pr op osition 4.2. By (1), S B C ( K ) ∗ ⊆ F ∗ is a Gruenhag e compact in the w ∗ -top ology , giving (3).  References 1. R. Devill e, G. 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