Farthest points of sets in uniformly convex banach spaces

LetS be a closed and bounded set in a uniformly convex Banach spaceX. It is shown that the set of all points inX which have a farthest point inS is dense. Letb(S) denote the set of all farthest points ofS, then a sufficient condition for to hold is thatX have the following property (I): Every closed and bounded convex set is the intersection of a family of closed balls.     

A banach space containing non-trivial limited sets but no non-trivial bounding sets

An example of a Banach spaceE is given with the following properties: Every bounding setA⊂E (i.e.f(A) is bounded for each holomorphic functionf:E →C) is relatively compact but there are relatively non-compact limited setsA (i.e.T(A) is relatively compact for each bounded linear mapT:E →c ₀).     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

Farthest points in reflexive locally uniformly rotund Banach spaces

IfS is a bounded and closed subset of a Banach spaceB, which is both reflexive and locally uniformly rotund, then, except on a set of first Baire category, the points inB have farthest points inS.     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

On well posedness of best simultaneous approximation problems in Banach spaces

The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

A superreflexive banach spaceX withL(X) admitting a homomorphism onto the Banach algebraC(βN)

A separable superreflexive Banach spaceX is constructed such that the Banach algebraL(X) of all continuous endomorphisms ofX admits a continuous homomorphism onto the Banach algebraC(βN) of all scalar valued functions on the Stone-Čech compacification of the positive integers with supremum norm. In particular: (i) the cardinality of the set of all linear multiplicative functionals onL(X) is equal to 2^c and (ii)X is not isomorphic to any finite Cartesian power of any Banach space.     

Injective Banach spaces of typeC(T)

A Banach spaceX is aP _λ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aP _λ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setT _Asuch thatC(T_A) is aP _λ?1-space.     

A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach space? _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element of? _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Norms with locally Lipschitzian derivatives

If a separable Banach spaceX admits a real valued function ф with bounded nonempty support, φ 艂 is locally Lipschitzian and if no subspace ofX is isomorphic toc _o, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX. This author's research supported in part by NSERC (Canada) Grant A7535.     

Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here: