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

ISBN-10: 9812560602

ISBN-13: 9789812560605

This quantity comprises 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 useful. for instance, Benyamini's is the single up to date survey of the intriguing and lively quarter of the type of Banach areas lower than uniformly non-stop maps whereas Gonzбlez's article is a pioneer creation 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**

17. G. Godefroy and N. J. , to appear. 18. G. Godefroy, N. J. Kalton and G. Lancien, Subspaces of co(N) and Lipschitz isomorphisms, GAFA 10 (2000), 798-820. 19. E. Gorelik, The uniform nonequivalence of L , and l,, Israel J. Math. 87 (1994), 1-8. 20. W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. SOC. 6 (1993), 851-874. 21. S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225251.

Put El = span{ej : j > N } . , y E Mid(%,-x,6) . For the other inclusion, assume that y satisfies 112 f yI( 5 1 + 6. Write y = zo z1, where zi E Ei,and chose a sign 0 = f l so that llx 0zoII 2 1. Then + + + 6IP L llx + @/(IP Hence llzlll 5 ( ( 1 + 6 ) P - 1)"' (1 + BZOI(P + ( ( 2 1( ( PL 1 + ( I Z l ( l P . = ((z 5 (2pb)'lP for 6 < 6(p). If the support of x is not finite, we use the argument above for a finitely supported vector which approximates x (where the degree of approximation depends only on 6).

T h e n X * & Y * i s a local Moreover, we can identify the space L l ( p , X ) of vector-valued integrable functions with X&L1 (p)and the space C ( K ,X ) of continuous vector Valued functions with X & C ( K ) 5 . From these facts, taking into account that 47 L 1 ( p ) *and L , ( p ) are C ( K )spaces and that C ( K ) *is a & ( p ) space, we obtain additional examples. 8. , the local properties. We refer to l5 for a good survey on ultrapowers of Banach spaces. See also '. Here we consider the ultrapowers of Banach spaces to give new examples of local dual spaces and to present some characterizations of the subspaces of X* which are local dual spaces of X .

### Advanced Courses of Mathematical Analysis I: Proceedings of the First International School by A. Aizpuru-Tomas, F. Leon-Saavedra

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