Subspaces of sequential spaces

It is proved that the space of continuous functions on the ordinary closed interval with the topology of pointwise convergence is not subsequential. In sequential spaces satisfying certain conditions, subspaces dense-in-themselves without convergent sequences are found; such subspaces are constructed in certain sequential compact spaces and semitopological groups. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 407–413, September, 1998. The author thanks the participants of Arkhangel'skii's seminar held at Moscow State University for useful discussions of this work.     

Ultrapowers of spaces of integrable functions with respect to a sequential vector measure and locally bounded sets of functions

Consider the space of functions that are integrable with respect to a vector measure. In this paper we deal with such spaces defined using a particular class of measures that we call sequential vector measures. We prove that these spaces always define complemented subspaces of their ultrapowers. We use this result to characterize increase properties of certain families of functions by mean of the related projection.     

When a compact (countably compact) set is closed. II

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T ₄,-spaces.     

Homogeneous and multilinear forms related to a theorem of Jordan and von Neumann

For vector spaces over certain algebraic fields, homogeneous functionals which are derivable from multilinear forms are characterized by an identity involving two-dimensional subspaces. The quadratic case is the “parallelogram law” which distinguishes Hilbert spaces among Banach spaces.     

Polar Subspaces and Automatic Maximality

This paper is about certain linear subspaces of Banach SN spaces (that is to say Banach spaces which have a symmetric nonexpansive linear map into their dual spaces). We apply our results to monotone linear subspaces of the product of a Banach space and its dual. In this paper, we establish several new results and also give improved proofs of some known ones in both the general and the special contexts.     

Remarks on the Existence of Unconditional Bases for Weighted Countably-Hilbert Spaces and Their Complemented Subspaces

We give versions of a criterion for existence of unconditional bases for countably-Hilbert spaces. As applications we obtain theorems on existence of unconditional bases for certain classes of countably-Hilbert function spaces and for their complemented subspaces under additional constraints on the space and the corresponding projections to the complemented subspaces. These classes include generalizations of power series spaces of finite type and Kothe spaces determined by Dragilev-type functions.     

On disjoint function systems and complemented subspaces

A sufficient condition for the complementability of subspaces generated by disjoint function systems in rearrangement invariant spaces is given. Orthogonal projections in L _p -spaces are extended to certain rearrangement invariant spaces. Applications to Lorentz spaces are given.     

Ritz-Galerkin approximations in blending function spaces

Summary This paper considers the theoretical development of finite dimensional bivariate blending function spaces and the problem of implementing the Ritz-Galerkin method in these approximation spaces. More specifically, the approximation theoretic methods of polynomial blending function interpolation and approximation developed in [2, 11–13] are extended to the general setting of L-splines, and these methods are then contrasted with familiar tensor product techniques in application of the Ritz-Galerkin method for approximately solving elliptic boundary value problems. The key to the application of blending function spaces in the Ritz-Galerkin method is the development of criteria which enable one to judiciously select from a nondenumerably infinite dimensional linear space of functions, certain finite dimensional subspaces which do not degrade the asymptotically high order approximation precision of the entire space. With these criteria for the selection of subspaces, we are able to derive a virtually unlimited number of new Ritz spaces which offer viable alternatives to the conventional tensor product piecewise polynomial spaces often employed. In fact, we shall see that tensor product spaces themselves are subspaces of blending function spaces; but these subspaces do not preserve the high order precision of the infinite dimensional parent space.Considerable attention is devoted to the analysis of several specific finite dimensional blending function spaces, solution of the discretized problems, choice of bases, ordering of unknowns, and concrete numerical examples. In addition, we extend these notations to boundary value problems defined on planar regions with curved boundaries.     

On countable choice and sequential spaces

Under the axiom of choice, every first countable space is a Fréchet‐Urysohn space. Although, in its absence even ? may fail to be a sequential space. Our goal in this paper is to discuss under which set‐theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ?, are classes of Fréchet‐Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in ? if and only if the axiom of countable choice holds for families of subsets of ?, and every metric space has a unique ‐completion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The isometry groups of the hamming spaces of periodic sequences

We consider the Hamming space of periodic (0, 1)-sequences and a continual family of its subspaces defined as direct limits of finite Hamming spaces. These subspaces form a complete lattice under inclusion which is isomorphic to the lattice of supernatural numbers. We explicitly describe the isometry groups of these spaces. This involves certain constructions similar to the hyperoctahedral groups but accounting for additional structures on the underlying sets.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Minimal vectors in arbitrary Banach spaces

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.     

Sur un théorème de J. L. Krivine concernant la caracterisation des classes d’espaces isomorphes a des espaces d’orlicz généralisés et des classes voisines

The classL ^p the classes λ-SL ^p of spaces λ isomorphic to subspaces ofL ^p spaces and λ-QL ^p of subquotients have been characterized in the literature by formulas of certain simple forms. A theorem of Krivine gives a general demonstration of these results in the framework of ψ normed spaces. In particular, characterizations of subspaces and subquotients of certain classes of generalized Orlicz spaces are obtained.      

The Eigenspaces of the Bose-Mesner-Algebras of the Association Schemes Corresponding to Projective Spaces and Polar Spaces

In an earlier paper 7, some properties of the eigenspaces of the Bose-Mesner-algebras of association schemes are figured out, leaving open the problem of determining the eigenspaces. In the present paper, these eigenspaces and the eigenvalues are determined for projective spaces and for polar spaces. This allows characterizations of certain sets of subspaces of these geometries.     

The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm

Coupled cell systems are networks of dynamical systems (the cells), where the links between the cells are described through the network structure, the coupled cell network. Synchrony subspaces are spaces defined in terms of equalities of certain cell coordinates that are flow-invariant for all coupled cell systems associated with a given network structure. The intersection of synchrony subspaces of a network is also a synchrony subspace of the network. It follows, then, that, given a coupled cell network, its set of synchrony subspaces, taking the inclusion partial order relation, forms a lattice. In this paper we show how to obtain the lattice of synchrony subspaces for a general network and present an algorithm that generates that lattice. We prove that this problem is reduced to obtain the lattice of synchrony subspaces for regular networks. For a regular network we obtain the lattice of synchrony subspaces based on the eigenvalue structure of the network adjacency matrix.     

Maximal inner spaces and Hankel operators on the Bergman space

The paper deals with two closely related questions about the Bergman space of the unit disk. First, we investigate a special class of invariant subspaces of the Bergman space, namely, invariant subspaces induced by certain Hankel operators. We show that such spaces always have the co-dimension 1 or 2 property; and we determine exactly when such a space has the co-dimension 1 property. Second, we introduce the notion of inner spaces in the Bergman space and give several characterizations of when an inner space is maximal.Research supported by the National Science Foundation     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.