RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 103 (1), 2009, pp. 87–90 An´ alisis Matem´ atico / Mathematical Analysis About an example of a Banach space not weakly K -analytic J. Ka ¸kol and M. L ´ opez Pellicer Abstract. Cascales, Ka ¸kol, Saxon proved [3] that in a large class G of locally convex spaces (lcs) (containing (LM)-spaces and (DF )-spaces) for a lcs E ∈ G the weak topology σ(E,E ′ ) of E has countable tightness iff its weak dual (E ′ ,σ(E ′ ,E)) is K-analytic. Applying examples of Pol [9] (and Kunen [12]) one gets that there exist Banach spaces C(X) over a compact scattered space X such that C(X) is not weakly K-analytic (even not weakly K-countably determined under (CH)) but the weak dual of C(X) has countable tightness. This provides also an example showing that (gDF )-spaces need not be in class G. Sobre un ejemplo de un espacio de Banach no d´ ebilmente K-anal´ ıtico Resumen. Cascales, Ka ¸kol, Saxon probaron en [3] que en la amplia clase G de espacios localmente convexos, que contienen a los espacios (LM) y a los espacios (DF ), se tiene que un espacio E ∈ G veri- fica que su topolog´ ıa d´ ebil σ(E,E ′ ) tiene tightness numerable si y s ´ olo si su dual d´ ebil (E ′ ,σ(E ′ ,E)) es K-analitico. Aplicando ejemplos de Pol [9] (y de Kunen [12]) se obtiene que existen compactos disemi- nados (scattered) X tales que el espacio de Banach C(X) no es d´ ebilmente K-analitico (y ni siquiera es d´ ebilmente K-numerablemente determinado, si se admite la hip´ otesis del continuo), pero el dual d´ ebil de C(X) tiene tightness numerable, lo que proporciona de exemplo de espacio (DF ) generalizado (espacio (gDF )) que no pertenece a la clase G. 1 Introduction Many concrete locally convex spaces (lcs) E are weakly realcompact, i.e. the weak topology σ(E,E ′ ) is realcompact. For example any lcs E whose strong dual is metrizable and separable has this property. This follows from (*): If (E,E ′ ) is a dual pair and σ(E,E ′ ) has countable tightness (i.e. for every A ⊂ E and x ∈ A there exists countable B ⊂ A such that x ∈ B), then σ(E ′ ,E) is realcompact. Indeed, by Corson [4], see also [17, p.137], it is enough to show that every linear functional f on E which is σ(E,E ′ )- continuous on each σ(E,E ′ )-closed separable vector subspace is continuous. Note that K := f −1 (0) is closed in σ(E,E ′ ): If y ∈ K, then there is countable D ⊂ K with y ∈ D. By assumption f | lin(D) is σ(E,E ′ )-continuous, so f (y) ∈ f (lin(D)) ⊂ lin f (K)= {0}. Hence y ∈ K, so f is continuous. The following is much less evident: Every Banach space with Corson property (C) is weakly real- compact,[4, 11]. It turns out that in a class G of lcs (containing all (LM )-spaces and (DF )-spaces) the converse to (*) holds even in a stronger form [3]. Proposition 1 If E ∈ G, then E σ := (E,σ(E,E ′ )) has countable tightness iff E ′ σ := (E ′ ,σ(E ′ ,E)) is K-analytic. Presentado por / Submitted by Dar´ ıo Maravall Casesnoves. Recibido / Received: 2 de marzo de 2009. Aceptado / Accepted: 4 de marzo de 2009. Palabras clave / Keywords: Banach C(X) space, compact scattered, countable tightness, K-analytic, K-countable determined, locally convex space, weak topology. Mathematics Subject Classifications: 46A03, 46E15, 54C30, 54H05. c  2009 Real Academia de Ciencias, Espa˜ na. 87