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

This quantity involves a suite of articles from specialists with a wealthy learn and academic event. The individuals of this quantity are: Y Benyamini, M Gonzбlez, V Mьller, S Reich, E Matouskova, A J Zaslavski and A R Palacios. every one in their paintings is valuable. for instance, Benyamini's is the one up to date survey of the fascinating and lively quarter 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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A. Yorke, Prevalence: A translation-invariant “almost every” on infinite dimensional spaces, Bull. Amer. Math. SOC. 27 (1992), 217-238. 23. B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: A n addendum, Bull. Amer. Math. 28 (1993), 306-307. 24. D. J. Ives and D. Preiss, Not too well differentiable Lipschitz isomorphisms, Israel J. Math. 115 (ZOOO), 343-353. 25. W. B. Johnson, J. Lindenstrauss and G. Schechtman, Banach spaces determined by their uniform structure, Geom. Funct. Anal. 6 (1996), 430-470.

As the set MI it is possible to take the set of all finite linear combinations of the basis vectors ei. Properties (ii) and (iii) follow from general results that will be discussed in the subsequent sections. The paper is organized as follows. In the following section we study regular orbits. Of particular interest are the orbits satisfying I(Tnzll 4 00. It is easy to see that if an operator T has such an orbit, then it has a nontrivial closed invariant subset {Tnz : n = 0 , 1 , . . }-. , the vectors with very irregular orbits.

In this case, x J * is a projection from X * onto x ( Y * ) . - In the following result we give “global” characterizations of the property of being a dual local. These characterizations are easier to deal with than the definition. However there is a disadvantage: the second dual X** appears in the statements. Theorem 5. (Global characterizations) For a subspace Z of X * , the fol- lowing statements are equivalent: - (1) Z is a local dual of X . (2) there is an isometric extension operator x : Z* R(X) 3 (3) there exists a norm-one projection P : X** N ( P ) = Z* and R ( P ) 2 X ; (4) there exists a norm-one projection Q : X*** R(Q) = 2’’ and N ( Q ) c X I .