Q spaces and Morrey spaces

We characterize functions in Morrey space by p-Carleson measures. We then reveal a simple relation between Q space and Morrey space, that is Q space can be viewed as a fractional integration of the Morrey space. Therefore, many results for Morrey space can be translated onto Q space. For example, we show that Q space is a dual space by identifying its predual.     

Some generalizations of locally and weakly locally uniformly convex space

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.     

COARSE AND UNIFORM EMBEDDINGS INTO REFLEXIVE SPACES

Answering an old problem in nonlinear theory, we show that c₀cannot be coarsely or uniformly embedded into a reflexive Banachspace, but that any stable metric space can be coarsely anduniformly embedded into a reflexive space. We also show thatcertain quasi-reflexive spaces (such as the James space) alsocannot be coarsely embedded into a reflexive space and thatthe unit ball of these spaces cannot be uniformly embedded intoa reflexive space. We give a necessary condition for a metricspace to be coarsely or uniformly embeddable in a uniformlyconvex space.     

A nonseparable Banach space not containing a subsymmetric basic sequence

We give an example of a nonseparable Banach space which does not contain a subsymmetric basic sequence. The space is the dual of a space constructed analogously to the James Tree space, using the Tsirelson space in place ofl ₂. Research partially supported by NSF Grant No. DMS-8403669.     

Frechet-valued martingales and stochastic integrals

Stochastic processes with values in a separable Frechet space whose a itinuous linear functional are real-valued square integrable martingales are investigated. The coordinate measures on the Fréchet space are obtained from cylinder set measures on a Hilbert space that is dense in the Fréchet space. Real-valued stochastic integrals are defined from the Fréchet-valued martingales using integrands from the topological dual of the aforementioned Hilbert space. An increasing process with values in the self adjoint operators on the Hilbert space plays a fundamental role in the definition of stochastic integrals. For Banach-valued Brownian motion the change of variables formula of K. Itô is generalized. A converse to the construction of the measures on the Fréchet space from cylinder set measures on a Hilbert space is also obtained.     

A characterization of reflexive spaces

We prove that a Banach space is reflexive if for every equivalent norm, the set of norm attaining functionals has non-empty norm-interior in the dual space. It is also proved that the set of norm attaining functionals on a Banach space that is not a Grothendieck space is not a w*-G _δ subset of the dual space.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

The dual of C ps (X)

This is a study of the dual space of continuous linear functionals on the function space C_ps(X) with a natural norm inherited from a larger Banach space. Here ps denotes the pseudocompact-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X. The lattice structure and completeness of this dual space have been studied. Since this dual space is inherently related to a space of measures, the measure-theoretic characterization of this dual space has been studied extensively. Due to this characterization, a special kind of topological space, called pz-space, has been studied. Finally the separability of this dual space has been studied.      

Extension of an abstract attainability problem with the use of the stone representation space

We consider an attainability problem in a complete metric space on values of an objective operator h. We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a nonempty family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.     

On generalized projective spaces operating on sets

A skewprojective space is a generalization of both groups and projective spaces. It is desarguesian if it is the space of infinity of a suitable skewaffine space. Especially a projective space is desarguesian in the sense cited above iff it is desarguesian in the usual sense. As a generalization of the well known fact that a proper subspace of a projective space is always desarguesian we establish a large class of skew-projective spaces also possessing this property.Dedicated to Günter Pickert on his 80^th birthday     

On paratopological vector spaces

We show that each first countable paratopological vector space X has a compatible translation invariant quasi-metric such that the open balls are convex whenever X is a pseudoconvex vector space. We introduce the notions of a right-bounded subset and of a right-precompact subset of a paratopological vector space X and prove that X is quasi-normable if and only if the origin has a convex and right-bounded neighborhood. Duality in this context is also discussed. Furthermore, it is shown that the bicompletion of any paratopological vector space (respectively, of any quasi-metric vector space) admits the structure of a paratopological vector space (respectively, of a quasi-metric vector space). Finally, paratopological vector spaces of finite dimension are considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Compact convex sets and complex convexity

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a one-dimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be affinely embedded into the spaceL ₀ of all measurable functions.     

The Weil–Petersson Kähler Form and Affine Foliations on Surfaces

The space of broken hyperbolic structures generalizes the usual Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations – or, equivalently, the space of projectivized affine foliations of a punctured surface – likewise generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations provide a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil–Petersson Kähler two-form to a corresponding two-form on the space of broken hyperbolic structures as well as Thurston's symplectic form to a corresponding two-form on the space of broken measured foliations, and we show that the former limits in an appropriate sense to the latter. The proof in sketch follows earlier work of the authors for measured foliations and depends upon techniques from decorated Teichmüller theory, which is also applied here to a further study of broken hyperbolic structures.     

Fréchetness of inverse limits and products

Let X be a Suslin-Borel set in a compact space. It is proved that X is either σ-scattered or contains a compact perfect set. If X is first countable, the result remains valid when X is a Suslin-Borel set in a Prohorov space. It is also proved that every first countable Prohorov space is a Baire space.     

Universal and homogeneous structures on the Urysohn and Gurarij spaces

Using Fraïssé theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, L-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm L on the Lispchitz-free space of the Urysohn space.Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a 1-complemented subspace. We construct the Gurarij space by the classical Fraïssé theoretic approach.     

Very smooth points of spaces of operators

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is anM-ideal in the space of bounded operators, a very smooth operatorT attains its norm at a unique vectorx (up to a constant multiple) andT(x) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon-Nikodym property. We give an example of a smooth Banach space without any very smooth points.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

Banach空间X的弱局部Zk性质

本文引入了Ｂａｎａｃｊ空间Ｘ的弱局部Ｚ_k性质和弱Ｚ_k性质，得出了：Ｂａｎａｃｈ空间Ｘ是Ｋ－ＮＵＣ空间的一个充分必要条件和Ｂａｎａｃｈ空间Ｘ是ＬＫ－ＮＵＣ空间的一个充分条件，并指出弱Ｚ_k性质蕴含弱Ｂａｎａｃｈ－Ｓａｋｓ性质．     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

On closed subspaces with Schauder bases in non-archimedean Fréchet spaces

The main purpose of this paper is to prove that a non-archimedean Fréchet space of countable type is normable (respectively nuclear; reflexive; a Montel space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Montel space). It is also shown that any Schauder basis in a non-normable non-archimedean Fréchet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Fréchet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Fréchet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.