ALGEBRA AND BANACH SPACES

Author: ROMERO MARTÍN RAÚL.
Year: 2004.
University: COMPLUTENSE DE MADRID [www.ucm.es].
Place of defense: FACULTAD DE CC. MATEMÁTICAS.
Place of preparation: FACULTAD DE CC. MATEMÁTICAS.

Summary: In this report, discusses methods of interpolation for more than two spaces banach focusing on the interpolation spaces orlicz with weights for this are extensions of the method of Gustavsson-Peetre the case of families of finite spaces banach. Another part of the report is devoted to the study of interpolation of compactness, establishing the best possible results from type Lions-Peetre for families of finite spaces Banach.Se also studying the validity of the results when working with other ideals of operators other than the ideal of compact operators.

Author: YÁÑEZ MURILLO DIEGO FRANCISCO.
Year: 2004.
University: EXTREMADURA [www.unex.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: This produces a new characterization "two-dimensional, real and isometric" spaces prehilbertianos; these is, in the spaces regulated whose rule is induced by a scalar product in the line which appeared in the works of Amir D. "Characterizations of inner product spaces ", Birkauser Verlag, Basel, 1986; type of" Equality of parallelogram "or" theorem Brunn, and Blaschke Kakutani. More specifically, set a real number r (or lesser r menor1), it is proven that an area regulated X (real or complex), S unit area, is prehilbertiano if and only if. It is also studying applications of this new characterization of spaces prehilbertianos: one consisting of using the new characterization above, obtain a new characterization of the spaces prehilbertianos using medium Fermat-Torricelli three points, in connection with the classical problem the median Fermat-Torricelli (S XVII), the other consisting of weakening assumptions result of a classic Gurarii and Sozonov (1968) concerning characterization spaces prehilbertianos.

Author: Rodríguez Ruiz José.
Year: 2005.
University: MURCIA [www.um.es].
Place of defense: Facultad de Matemáticas.
Place of preparation: Facultad de Matemáticas.

Summary: This dissertation is part of the theory of integration of functions with values in Banach spaces. The report consists of five chapters and an introduction, the first one of ancillary. In Chapter 2 we look at the comprehensive Birkhoff of functions defined in a space of probability values in a Banach space. Caracterizamos completely integrabilidad Birkhoff a feature vector in terms of real functions of the family formed by the compositions of the role with the elements of the ball dual space. In this sense, the notion that appears associated with integrabilidad Birkhoff is the so-called owned Bourgain of a family of real functions. As application, replace integrabilidad Pettis by integrabilidad Birkhoff well known in the characterization of Banach spaces without copies of l1 as those areas which have dual ownership of Radon - Nikodym weak. In particular, this allows us to resolve, in the case of dual spaces, a problem proposed by Fremlin concerning representation of vector measures like indefinite integrals of functions integrated McShane. In Chapter 3 we consider various theories of integrating functions vector from measures vector, including S *- integral Dobrakov (which is the extension of the integral of Birkhoff in this context) and the widespread nature of the comprehensive McShane . Our main theorem ensures that every function is integrable S *- Integrable McShane, the reciprocal is true for functions strongly measurable. We tried both S *- integrable functions such as integrated McShane can approximate arbitrarily by simple functions in the standard given by the semivariación the indefinite integral. In Chapter 4 we study and compare various methods of integration (Debreu, Birkhoff and Pettis) multi-funciones defined in a space of probability values in the family of all subsets (not empty) and slightly convex compact space Banach separable. For example, we tested the notion of integrabilidad Birkhoff is intermediate (strictly) between the other two. Finally, in Chapter 5 relate integration vector with the theory of operators absolutely sumantes, considering how far the composition with one operator improves the properties of integrabilidad a feature vector.

Author: Avilés López Antonio.
Year: 2005.
University: MURCIA [www.um.es].
Place of defense: Facultad de Matemáticas.
Place of preparation: Facultad de Matemáticas.

Summary: The thesis is divided into four chapters. The first of these is devoted to compact Eberlein uniforms, ie weak compact subsets of Hilbert spaces. It proves that the ball of a Hilbert space in the weak topology is continuous image of a product numerable of compactificaciones by a discrete set, which combined with results Murray Bell involves the blinking of open ball that meets various properties type Ramsey. It also builds a rule in the Hilbert space is not separable different from the norm hilbertiana but equivalent thereto whose ball into the topology weak verica these properties Ramsey and therefore is not homeomorfo the ball in the standard hilbertiana. Another example of compact Eberlein uniform is the family compact subsets of natural No less than cardinality of a set, as well as to areas that are expressed as products numerables of such compacts. It also looks at the first chapter topological sorting and classifying spaces C (K) of K compact which are products numerables as described above. In the second chapter, is studied in the invariant center of a Banach space given by the smaller number of subassemblies needed to generate weak compact space. It explores the relationship of this invariant with others as the number of Lindelof or indexes K-analiticidad and K-determinación, revealing among other outcomes that exist Banach spaces weakly Lindelof generated by a certain number of arbitrarily high weak compact, while all space weakly numerablemente determined is generated by a continuing weak compact. It also explores the relationship between the number of weak compact space and generate a subspace, and the relationship with the density. For example, a subspace of a space weakly compactamente generated is always generated by d weak compact, and if this subspace is a density less than b, then it is also weakly compactamente generated. In chapter 3 are considered Asplund generated Banach spaces and compact Radon - Nikodym introduced generalizations in terms of cardinal invariant similar to those of the previous chapter. Notable progress on the issue of the image of the continuous compact Radon - Nikodym. It proves that different superclasses of such compact defined by several authors are all equal to the so-called compact almost Radon - Nikodym. It shows that a compact Radon - Nikodym that is almost completely sorted or having weight less than b is a compact Radon - Nikodym. Finally, the last chapter discusses properties renormamiento. They are used spaces James JT about different types of trees T to give examples of Banach spaces that do not contain l1 but which does not allow dual standards equivalent strictly convex or Kadec. There is a full characterization of the tree T for which JT is weakly compactamente generated and has dual standard Kadec (or URL) as an example of space JT weakly Lindelof determined which lacks both dual standard Kadec as standard strictly convex.