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Let (X, · ) be a (real) Banach space. We refer to [38] or [28] as some introduction to the general theory of Banach spaces. Note that, as usual in the case, all the results we discuss here remain valid for complex scalars with possibly different constants. Let I be a countable set with possibly some ordering we refer to whenever considering convergence with respect to elements of I (wich will be denoted by limi→∞). Definition 1 We say that countable system of vectors is biorthogonal if for i, j ∈ I we have

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

In Chapter 16 we give applications to various topics of Banach space theory, the most spectacular of which is a sharpened version of a celebrated result of Jean Bourgain on the Λ p problem.

In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.

In this paper we study the Hardy–Lorentz spaces for Banach space valued martingales. The dual spaces are characterized and several martingale inequalities are established. The proofs mainly depend on the classical tool of atomic decompositions. As usual, these conclusions are closely related to the geometrical properties of the underlying Banach spaces.

We exhibit a new class of Banach spaces Y such that the pair ( c 0 , Y ) has the Bishop–Phelps–Bollobas property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.

In this study, we introduce the norm-attaining property for a dual pair of Banach spaces. We use this property to determine a quotient Banach space as a dual space. We also apply this property to obtain another proof of the James's characterization theorem regarding the reflexivity of a separable Banach space. In addition, the norm-attaining property of a Fourier space is studied.

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

The purpose of this paper is to introduce and study the split equality variational inclusion problems in the setting of Banach spaces. For solving this kind of problems, some new iterative algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in the paper to study the split equality feasibility problems in Banach spaces and the split equality equilibrium problem in Banach spaces. The results presented in the paper are new.

Motivated by recent applications of weighted norm inequalities to maximal regularity of first and second order Cauchy problems, we study real interpolation spaces on the basis of general Banach function spaces and, in particular , weighted rearrangement invariant Banach function spaces. We show equivalence of the trace method and the K-method, identify real interpolation spaces between a Banach space and the domain of a sectorial operator, and reprove an extension of Dore's theorem on the boundedness of H ∞-functional calculus to this general setting.

In this paper we study totally bounded sets in Banach function spaces (BFS), from which we characterize compact sets (via Hausdorff criterion) in some non-standard function spaces which fall under the umbrella of BFS. We obtain a Riesz–Kolmogorov compactness theorem for the grand variable exponent Lebesgue spaces.

In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the main results about quasidensity, and give some simple limiting examples in Hilbert spaces, reflexive Banach spaces, and nonreflexive Banach spaces.

International audience; The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the so-called Karhunen–Loève expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standard Wiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.

We give a characterization of weakly subgaussian random elements that are γ-subgaussian in infinite-dimensional Banach and Hilbert spaces.

The purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.

We discuss the Banach space structure of the fractional order weighted Fock-Sobolev spaces F , , mainly include giving some growth estimates for Fock-Sobolev functions and approximating them by a sequence of ‘nice’ functions in two different ways.

In this chapter, differentiation and integration of operators defined on a Banach space into another Banach space are introduced. Basic concepts of distribution theory and Sobolev spaces are discussed, both concepts play very significant role in the theory of partial differential equations. A lucid presentation of these two topics is given.

The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.

We present simple proofs that spaces of homogeneous polynomials on Lp[0, 1] and lp provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

In this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F-valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach-Saks property, separable Banach spaces not containing c0, Banach spaces not containing c0 or l1 and Asplund spaces not containing c0.

A Banach space X is called subprojective if any of its infinite dimensional subspaces contains a further infinite dimensional subspace complemented in X. This paper is devoted to systematic study of subprojectivity. We examine the stability of subprojectivity of Banach spaces under various operations, such as direct or twisted sums, tensor products, and forming spaces of operators. Along the way, we obtain new classes of subprojective spaces.

In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.

We extend some results about sampling of entire functions of exponential type to Banach spaces. By using the generator D of a one-parameter group etD of isometries of a Banach space E we introduce Bernstein sub-spaces Bσ(D), σ > 0, of vectors f in E for which trajectories etD f are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for etD f with f ∊ Bσ(D) to known sampling results for regular functions of exponential type σ.

In this paper it is shown that: (1) If every weak⁎ hyperplane of X ⁎ is approximatively compact, then (a) X is an Asplund space; (b) X ⁎ has the Radon–Nikodym property. (2) Criteria for approximative compactness of every weakly⁎ hyperplane of Orlicz–Bochner function spaces equipped with the Orlicz norm are given. (3) If X has a Frechet differentiable norm, then (a) Orlicz–Bochner function spaces L M 0 ( X ⁎ ) have the Radon–Nikodym property if and only if M ∈ Δ 2 ; (b) Orlicz–Bochner function spaces E N ( X ) are Asplund spaces if and only if M ∈ Δ 2 . (4) We give an important application of approximative compactness to the theory of generalized inverses for operators between Banach spaces and Orlicz–Bochner function spaces.

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck’s approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces.