Infinite-dimensional Real Banach Space Research Articles

A hyperbolicity criterion for a class of diffeomorphisms of an infinite-dimensional torus

On an infinite-dimensional torus , where is an infinite-dimensional real Banach space and is an abstract integer lattice, a special class of diffeomorphisms is considered. It consists of the maps equal to sums of invertible bounded linear operators preserving and -smooth periodic additives. Necessary and sufficient conditions ensuring that such maps are hyperbolic (that is, are Anosov diffeomorphisms) are obtained. Bibliography: 15 titles.

Cross-sections of solution funnels

Let X be a separable infinite dimensional real Banach space. We denote by F(X) the class of continuous functions f:R×X→X such that the ODE u′=f(t,u),u(t0)=x,t0∈R,x∈X, has a global solution for any initial condition. Our main result states that A⊂X is the cross-section of a solution funnel of the ODE u′=f(t,u),u(0)=0, for some f∈F(X), if and only if A is an analytic set.

The image of a closed convex set under a Fredholm operator

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {Dn:n∈N}, a non-decreasing family of non empty, closed, convex and bounded sets such that Dn+Dn+2⊆2Dn+1 for every n∈N.The second part of this article proves that in every infinite dimensional real Banach space there is a convex set which can be expressed as the union of countably many closed sets, but not as the union of countably many closed and convex sets. Accordingly, every infinite dimensional real Banach space contains a convex Fσ set which is not the image of a closed convex set under a Fredholm operator.

On rearrangement theorems in Banach spaces

Abstract. It is shown that every infinite-dimensional real Banach space X contains a sequence ( x n ) n ∈ ℕ $(x_n)_{n\in \mathbb {N}}$ with the following properties: (a) Some subsequence of ( ∑ k = 1 n x k ) n ∈ ℕ $(\sum _{k=1}^nx_k)_{n\in \mathbb {N}}$ converges in X and sup n ∈ ℕ ∥ ∑ k = 1 n x k ∥ ≤ 1 $\sup _{n\in \mathbb {N}}\Vert \sum _{k=1}^nx_k\Vert \le 1$ ; (b) ∑ k = 1 ∞ ∥ x k ∥ p < ∞ $\sum _{k=1}^{\infty }\Vert x_k\Vert ^p<\infty $ for every p ∈ ] 2 , + ∞ [ $p\in {]2,+\infty [}$ ; (c) for any permutation π : ℕ → ℕ $\pi :\mathbb {N}\rightarrow \mathbb {N}$ and any sequence ( θ n ) n ∈ ℕ $(\theta _n)_{n\in \mathbb {N}}$ with θ n ∈ { - 1 , 1 } $\theta _n\in \lbrace -1,1\rbrace $ , n = 1 , 2 , ... $n=1,2,\ldots $ , the series ∑ k = 1 ∞ θ k x π ( k ) $\sum _{k=1}^{\infty }\theta _k x_{\pi (k)}$ diverges in X. This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

On real analytic functions of unbounded type

In this paper we prove several results on the existence of analytic functions on an infinite dimensional real Banach space which are bounded on some given collection of open sets and unbounded on others. In addition, we also obtain results on the density of some subsets of the space of all analytic functions for natural locally convex topologies on this space.

On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x ′ = f ( x ) , x ( 0 ) = x 0 , where f : X → X is Lipschitz, as being of three types I–III. We denote by S X the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x 0 ∈ X . We say that S ∈ S X is of type I if there exists a Lipschitz function f and a solution x such that S = Ω ( x ) and { x ( t ) : t ⩾ 0 } ∩ S = ∅ . We say that S ∈ S X is of type II if it has non-empty interior. We say that S ∈ S X is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S = Ω ( x ) it holds { x ( t ) : t ⩾ 0 } ⊂ S . Our main results are the following: S is a type I set in S X if and only if S is a closed and separable subset of the topological boundary of an open and connected set U ⊂ X . Suppose that there exists an open separable and connected set U ⊂ X such that S = U ¯ , then S is a type II set in S X . Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen C k -smooth whenever the underlying Banach space is C k -smooth.

Trigonometric and Rademacher measures of nowhere finite variation

Let X X be an infinite dimensional real Banach space. It was proved by E. Thomas and soon thereafter by L. Janicka and N. J. Kalton that there always exists a measure μ \mu into X X with relatively norm-compact range such that its variation measure assumes the value ∞ \infty on every non-null set. Such measures have been called “measures of nowhere finite variation” by K. M. Garg and the author, who as well as L. Drewnowski and Z. Lipecki have done related investigations. We give some “concrete” examples of such μ \mu ’s in the l p l^p spaces defined using the (real) trigonometric system ( t n ) (t_n) and the Rademacher system ( r n ) (r_n) illustrating similarities and some differences. We also look at the extensibility of the integration map of these μ \mu ’s. As an application of the trigonometric example, we have the probably known result: For every p ≥ 1 p\ge 1 , the function ( Σ ( | t n ( t ) | p ) / n ) (\Sigma (| t_n (t) | ^p ) / n ) is unbounded on every set A A with positive measure.

Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C ( K ) examples due to Plebanek [G. Plebanek, A construction of a Banach space C ( K ) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.

Real Banach Spaces with Numerical Index 1

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l 1 . It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

Bounded Real-Analytic Submanifolds of Banach Spaces

It is shown that, in contrast to the complex case, every paracompact finite-dimensional real-analytic manifold can be embedded as a bounded and closed real-analytic submanifold of every infinite-dimensional real Banach space.

On a vector space construction by Hausdorff

Initroduction. Some years ago Hausdorff, using Hamel bases, showed that any infinite dimensional real Banach space contained a second category linear subspace that was not complete under any equivalent norm [5]. It is shown below that a slight abstraction of his construction leads to the following: (A) a multiple offender example (Theorem 3), one that combines several pathological features which have been separately discussed in the literature;2 (B) the partitioning of any infinite dimensional second category real linear topological space into a continuum of pairwise disjoint linear manifolds each of which is everywhere dense in the space (a result comparable to the partitionings of certain linear topological spaces into disjoint everywhere dense convex sets by Tukey [15 ] and Klee [8 ]); (C) some miscellaneous observations, including the fact that in some recent topological group theorems certain hypotheses cannot be dropped. To be precise concerning (A), we shall establish the following theorem, in which En is n-dimensional real Euclidean vector space and L. represents, for any element c of En, the set of rational multiples of c. Denoting complements by primes, a set E in En will be called m-thick if both E and E' have zero interior measure; when E is in a topological space X it is c-thick if each of the sets E and E' contains no Baire subsets of X other than first category sets.