Resumen: We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space C-2. This answers a question posed by R. Grzaslewicz and K. John , who solved the corresponding problem for the real Hilbert space R-2. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.
: Trilinear form; norm; Hilbert-Schmidt norm; extreme point; extreme point of unit ball
Resumen: In this note we prove that if a differentiable function oscillates between yÂ« and Â« on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less thanÂ« . This kind of approximate Rolleâ€™s theorem is interesting because an exact Rolleâ€™s
theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolleâ€™s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR).
Palabras clave: Rolleâ€™s theorem in infinite-dimensional Banach spaces; Approximate Rolleâ€™s theorem; Continuous norm whose dual norm is locally uniformly rotund; C1 bump function
Resumen: The authors consider the space of bounded holomorphic functions on the open unit ball of a Banach space with the natural analogue of the strict topology Î² for the one- (or finite-)dimensional case. This can be defined by means of weighted seminorms or as the mixed topology (in the sense of Wiweger) of the supremum norm and the topology of uniform convergence
on subsets of the unit ball which are bounded away from its complement. The natural analogues of some elementary properties of the one-dimensional case are obtained. In a second section, some properties of Hâˆž as a topological algebra are discussed. In particular, the spectrum is identified (under some rather restrictive conditions on the Banach space) and this is used to obtain a representation for strictly continuous homomorphisms between such Hâˆž-algebras
Palabras clave: Space of bounded holomorphic functions on the open unit ball of a Banach space; Strict topology; Approximation by polynomials; Continuous homomorphisms
Resumen: Let H be a two-dimensional complex Hilbert space and P(H-3) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B-P(3H), from which we deduce that the unit sphere of P(H-3) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B-P(3H) remains
extreme as considered as an element of B-L(3H). Finally we make a few remarks about the geometry of the unit ball of the predual of P(H-3) and give a characterization of its smooth points.
Resumen: Let X be an infinite-dimensional complex Banach space. Very recently, several results on the existence of entire functions on X bounded on a given ball B(1) subset of X and unbounded on another given ball B(2) subset of X have been obtained. In this paper we consider the problem of finding entire functions which are uniformly bounded on
a collection of balls and unbounded on the balls of some other collection.
Resumen: Two main results presented by the authors include a mean-value inequality for a class of Gateaux subdifferentiable functions and a subdifferential Rolleâ€™s theorem in a Banach space. For the second part, if a (Gateaux/Fréchet)subdifferentiable function f is bounded by " on the boundary of a unit ball, then there exists a Gateaux/Fréchet) subgradient of f at an interior
point of the ball which is bounded by 2".
Palabras clave: GÃ¢teaux subdifferential; Mean value inequality theorems; Subdifferential approximate Rolle's theorem; Fréchet subdifferential
Resumen: El sector de la automoción ha impulsado numerosos desarrollos en el campo de los materiales, específicamente aceros, para disminuir el peso en sus estructuras y, por tanto, el consumo de combustible y las emisiones de CO2 a la atmósfera. Uno de estos nuevos aceros que ha despertado mayor interés es el acero dual-phase, perteneciente a la familia de aceros avanzados de alta resistencia (AHSS, Advanced High Strength Steels), debido a su composición
de ferrita y martensita la cual otorga a este acero una elevada resistencia con una excelente ductilidad. Sin embargo, los aceros dual-phase presentan cierta dificultad a la hora de someterse a un proceso de conformado en frío como es el perfilado de geometría variable, proceso usual para la fabricación de piezas estructurales de los automóviles. En el presente trabajo se ha llevado a cabo una extensa caracterización microestructural (mediante
microscopía óptica, microscopía electrónica de barrido y utilizando la técnica de electrones retrodispersados (EBSD)), mecánica (mediante ensayos de ultramicrodureza) y tribológica (mediante ensayos de ball-on-disc) con la finalidad de optimizar los procesos de conformado en frio de aceros dual-phase. [ABSTRACT] The automotive industry has prompted many developments in the field of materials,
particularly steels, to reduce the weight in their structures and, therefore, to reduce fuel consumption and CO2 emissions during the manufacturing of car bodies. One of these new advanced high strength steels (AHSS) is the dual-phase steel, that due to its composition of ferrite and martensite phases provide a high strength with excellent ductility. However, dualphase steels present some difficulty when they are subjected to a cold forming process such as variable geometry roll-forming, that is a usual process for the manufacture of structural parts of automobiles. The present work shows the extensive microstructural characterization (by optical microscopy, scanning electron microscopy and using backscattered electron technique (EBSD)), mechanical (by ultramicrohardness tests) and tribological (by ball-on-disc test) carried out in order to optimize the cold forming processes of dual-phase steels.