Banach Spaces

Remark 1. Hypothesis 2 is satisfied for example if supp(µ) is bounded, and this happens when µ is the law of a bounded random vector. This assumption used to be standard in the study of probability in Banach spaces. Kuelbs, however (see [11]), was able to construct a dominating point assuming that Hypothesis 2 holds only for some t > 0 instead of for every t. His result allows the proof of sharp asymptotics for the Large Deviations Principle (LDP) with minimal assumptions. In the discussion below we keep Hypothesis 2 because it gives us exponential tightness.

Throughout, E, G, and F will be complex Banach spaces, E a subspace of G. The question which gave rise to the material reviewed here was first posed by Dineen [D1] in relation to holomorphic completeness and in the context of more general locally convex spaces. It is the following: when can a continuous homogeneous polynomial P : E −→ F be extended to P : G −→ F ? Several answers—none complete—have been given for varying hypotheses on the spaces and the polynomials involved, and several applications have appeared. We attempt here to unify the existing approaches to the problem, and to point out the common ingredients in the solutions.

convex minimization problems on Banach spaces in an abstract setting. In section 4, we introduce and analyze the progressive PGD (with or without updates) and we provide some general convergence results. While working on this paper, the authors became aware of the work [7], which provides a convergence proof for the purely progressive PGD when working on tensor Hilbert spaces. The present paper can be seen as an extension of the results of [7] to the more general framework of tensor Banach spaces and to a larger family of PGDs, including updating strategies and a general selection of tensor subsets S 1 . In section 5, we present some classical examples of applications of

Let Ω ⊆ R n , n ∈ {2, 3}, be a given bounded domain with polyhedral boundary Γ, and let ν be the outward unit normal vector on Γ . Standard notation will be adopted for Lebesgue spaces L p (Ω) and Sobolev spaces W s,p (Ω), with s ∈ R and p > 1, whose corresponding norms, either for the scalar, vectorial, or tensorial case, are denoted by k·k 0,p;Ω and k·k s,p;Ω , re-

Wei, Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral projections, Appl.. Robles, J.Y Velez-Cerrada, Characterizations of a class of matrices [r]

This section is devoted to polynomials on Banach spaces; multilinear map- pings, tensor products, restrictions to finite dimensional spaces and differential calculus may all be used to define polynomials over infinite dimensional spaces. We adopt an approach to them using multilinear mappings. Polynomials are used to eventually define analytic functions by means of series expansions analo- gously to the one dimensional case. We will also recall the polynomial topologies τ(E , P( n E )) and τ (E, P(E )), which will be used in Chapter 3. References about polynomials on Banach spaces can be found in [Din99], [Muj86] y [Gam94].

The reader can find a review on the classical results on the Daugavet property in [16]; for the case of Banach lattices of functions, which is particularly important for this paper, see [1, 2] and the references therein. The following generalization of the notion of Daugavet property will be used. Following [4, Definition 1.2], we say that a continuous linear operator G : X → Y between Banach spaces is a Daugavet center if kG + T k = kGk + kT k is fulfilled for every rank-1 operator T : X → Y . For this notion and the main properties which are necessary for this paper, see also [3, 4].

In this chapter, we define and study linear maps between Banach spaces. Along the way, we introduce some definitions and results that play an important role in our study of these maps. First, we establish some basics concepts such as boundedness, closedness and invertibility of linear operators, as well as spectrum, resolvent set and resolvent of a linear operator on Banach spaces. In the second section, we ob- tain other results applying these concepts to linear operators on Hilbert spaces and we define selfadjoint operators and orthogonal projections. Finally, we study the spectral properties of a selfadjoint operator and some properties of the orthogonal projections and their action on Hilbert spaces.

Our aim here is to obtain some results on unbounded analytic functions on some open subsets of infinite dimensional real Banach spaces. The proofs, which need some particular techniques, will follow some ideas used in the proofs of similar re- sults recently obtained in the complex case, most of them by the authors. See [1-3] and [8].

Jimknez-Melado, An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions of difference equations in Banach spaces, J.. Schmeidel, [r]

Let S and T be Hausdor topological spaces and X and Y separable Banach spaces. If µ and ν are Radon measures dened in bor(S) and bor(T ) with values in X and Y respectively, we will prove the existence of a unique Radon measure λ in bor(S × T ) with values in the injective tensorial product of X and Y . This measure is such that for any A in bor(S) and B in bor(T ) , λ(A × B) coincides with tensorial product of µ(A) and ν(B) . This result generalizes a similar one proved by M. Gómez and G. Restrepo in 2009.

The concept of bilinear isometry can be naturally extended to the context of spaces of vector-valued continuous functions. Examples of bilinear isometries defined on these spaces can be found, for instance, in [7, Proposition 5.2], where the author provide certain compact spaces X and Banach spaces E for which there exists a bilinear isometry T : C(X, E) × C(X, E) −→ C(Y, E).

A continuaci´on mostraremos algunos conceptos y resultados que son necesarios para el desarrollo de este art´ıculo desarrollados por Bartle en [1] y de una manera ampliada por Posada en [6]. Sean (Ω, A ) un espacio de medici´ on y X un espacio de Banach. Una funci´on ν : A → X es una medida vectorial contablemente aditiva (σ-aditiva) si para toda sucesi´on disjunta (A k ) k ∈N en A , la sucesi´ on de sumas

In Section 3 we deal with Banach Jordan pairs and prove the main results (i), (ii) and (iii) of the paper. A key point in the proof of these results is the fact that the small radical of a Noetherian Banach Jordan pair is closed which, together with Lemma 2·6, proves via the Baire Category Theorem that the small radical is finite dimensional. This reduces the study of a Noetherian Banach Jordan pair V to the case that V is nondegenerate. The proof of (ii) lies in two fundamental facts: the abundance of uniform elements in Noetherian Jordan pairs and Lemma 3·5 that asserts that Noetherian Banach Jordan domains J are division Jordan algebras (the complex field in the complex case). As a consequence of (ii), we have that the Jacobson radical of a Noetherian Banach Jordan pair coincides with the small radical and therefore it is finite dimensional. Finally, (iii) follows from (ii) and Loos’ classical list of simple Jordan pairs of finite capacity, previously refined according to an idea of McCrimmon [19] in order to identify the centroids.

Los resultados que presentamos en este cap´ıtulo son ampliamente conocidos y han sido tomados de [13], [1], [7] y [3]. Nuestro inter´ es principal es probar una caracteri- zaci´ on (debida a S. Banach y M. A. Zarecki) de la clase de funciones absolutamente continuas definidas en un intervalo [ a, b ] y con valores reales. Uno de los elementos m´ as interesantes de la prueba que presentamos es el uso de la funci´ on indicatriz de Banach.

Desde el punto de vista hist´ orico, los primeros Teoremas del Punto Fijo surgieron en el contexto de demostrar la existencia de soluciones de ecuaciones diferencia- les, ecuaciones integrales, etc. Poincar´ e, Birkhoff, Picard, Brouwer, Banach, entre otros matem´ aticos, publicaron art´ıculos en referencia al tema. Desde entonces, la Teor´ıa del Punto Fijo contin´ ua desarroll´ andose tanto te´ oricamente como desde el punto de vista de las aplicaciones. Consideramos que, esquem´ aticamente, la Teor´ıa del Punto Fijo tiene tres grandes ramas de estudio: M´ etrico, Top´ ologico y Conjuntista. En cada una de estas ramas se tienen teoremas muy importantes; desde el punto de vista m´ etrico se tienen el Teorema del Punto fijo de Banach, Caristi, Kannan [15], etc; desde el punto de vista top´ ologico: el Teorema de Brou- wer [6], Teorema de Schauder [6], Teorema de Kakutani, Teorema de Schaeffer [6], etc, y desde el punto de vista conjuntista: el Teorema de Knaster-Tarski [25], Teorema de Tarski [25], etc.

This definition was rewritten in [6] considering only annihilation operators prov- ing they are enough to define the Hermite-Sobolev space. In the same work, the authors deal with weighted Sobolev spaces for weights in the Muckenhoupt class A p , defined for 1 < p < ∞ , as the set of weights (non-negative and locally integrable

Consequently, th e results of §4 can be used to generalize some known res u l t s and applica­ tions for matrix-valued functions over finite dimensional vector spaces ( f o r the latter, see e.g. [ FGP )) . The authors express their gratitude to Professors G. Corach and H. Porta for informative discussions on this subject.

Pasamos ahora a tratar el problema de la inversibilidad generalizada. En primer lugar, hemos de remarcar una propiedad que utilizaremos de ahora en adelante. Si una apliaci´ on T preserva fuertemente la inversibilidad generaliza- da y a ∈ A es un elemento inversible tal que T (a) es inversible, entonces es de f´ acil comprobaci´ on que T (a −1 ) = (T (a)) −1 . Utilizando esta propiedad, Boudi y Mbekhta lograron adaptar la demostraci´ on del teorema anterior al caso de inversibilidad generalizada. Sus resultados principales fueron los siguientes: Teorema 2.2.3 Sean A y B ´ algebras de Banach unitales y T : A → B una aplicaci´ on unital y aditiva. Entonces T preserva fuertemente la inversibilidad generalizada si, y solamente si, T es un homomorfismo de Jordan.

En este trabajo tesis estudiamos el problema de Banach generali- zado el cual es un problema geom´etrico y topol ´ogico. Inicialmente estudiamos algunos resultados b ´asicos sobre haces fibrados, inclu- yendo las definiciones b ´asicas y el teorema de estructura. Presen- tamos tambien algunos ejemplos de haces fibrados y seguido de ello estudiamos los haces vectoriales y haces vectoriales euclidia- nos.Luego revisamos el articulo del Dr. Luis Montejano Secciones homot´ eticas ver [i] . Finalmente estudiamos la soluci ´on del pro- blema de Banach,haciendo uso de los resultados trabajados en el seminario de geometr´ıa y topolog´ıa damos una soluci ´on elegante y formal..