By A. Aizpuru-Tomas, F. Leon-Saavedra

This quantity includes a suite of articles from specialists with a wealthy study and academic adventure. The members of this quantity are: Y Benyamini, M Gonzбlez, V Mьller, S Reich, E Matouskova, A J Zaslavski and A R Palacios. each one in their paintings is beneficial. for instance, Benyamini's is the one up-to-date survey of the intriguing and lively zone of the type of Banach areas lower than uniformly non-stop maps whereas Gonzбlez's article is a pioneer advent to the speculation of neighborhood duality for Banach areas.

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**Extra info for Advanced Courses of Mathematical Analysis I: Proceedings of the First International School**

**Example text**

A more precise statement is given by the following theorem. Recall that a subset M c X is called residual if its complement X \ M is of the first category. Equivalently, M is residual if and only if it contains a dense G6 subset. 2. Let T E B ( X ) and let (a,) be a sequence of positive numa, = 0 . Then the set of all points x E X with the bers such that property that llTnxll 2 a,llTnll f o r infinitely many n is residual. Proof. The statement is trivial if T is nilpotent. In the following we assume that T n # 0 for all n.

We refer to l 1 for more details. It is not difficult to show that this subspace 2 is isometric to the space C(A) of the continuous functions on the Cantor set A in [0,1]. P. , a sequence of finite rank operators T, acting on X so that lim IIT,a: - a:((= 0 for all 5 E X , n+m 44 (b) lim llTnll = 1 and n-co ( c ) TnTk = TkTn = Tmin{k,n); Using this fact we can prove the following result. , and let (T,) be a commuting 1-approximating sequence o n X . P. Theorem 12. 15 in Remark 13. Let X be a space with a monotone basis.

CLAIM:There exists an integer n such that Pn satisfies the properties of the map L associated to F , G and E in Theorem 3. Indeed, taking into account that the unit ball of a finite dimensional space is compact, (a) follows from (l),(b’) follows from (2) and (c’) follows from (3). Although the definition of local dual space is “local”, we shall see in Theorem 5 that it is possible to give “global” characterizations. The following result will be useful. Proposition 4. Let 2 be a local dual of X .