Polyhedrality in pieces

The aim of this paper is to present two tools,  and , that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere _SX$S_X$ or of certain subsets of the dual ball _BX?$B_{X^{?}}$ of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K)$C(K)$ spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the final section.     

Topological Properties of the Continuous Function Spaces on Some Ordered Compacta

Some new classes of compacta K are considered for which C(K) endowed with the pointwise topology has a countable cover by sets of small local norm-diameter.     

Approximation of norms on Banach spaces

Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on $c_0(\mathrm{\Gamma })$ can be approximated uniformly on bounded sets by polyhedral norms and $C^{\infty }$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the ‘discrete’ Lorentz spaces $d(w,1,\mathrm{\Gamma })$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number α, there exists a scattered compact space K having Cantor–Bendixson height at least α, such that every equivalent norm on $C(K)$ can be approximated as above.     

Polyhedral norms on non-separable Banach spaces

We prove the existence of equivalent polyhedral norms on a number of classes of non-separable spaces, the majority of which being of the form C(K). In particular, we obtain a complete characterization of those trees T, such that C₀(T) admits an equivalent polyhedral norm.     

Bidual renormings of Banach spaces

Given a separable Banach space X with no isomorphic copies of ℓ₁ and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidual of a Banach space with a shrinking unconditional Schauder basis and to the bidual of the James space.     

Structural elucidation, optical, magnetic and nonlinear optical properties of oxystyryl dyes

Structure, magnetic and optical properties of tetraphenylborate salts of 2,5-[1-methyl-4-[2-(4-hydroxyphenyl)ethenyl]piridinium]-propane and butane are performed in gas and condense phase by means of solution and solid-state conventional and linear-polarized IR-spectroscopy of oriented colloids in nematic liquid crystal suspension, UV-vis and fluorescence methods, HPLC tandem ESI mess spectrometry (MS/MS), (1)H, (13)C and (1)H-(1)H COSY NMR, TGV and DSC methods. Quantum chemical DFT calculations are used for performing of the structures, optical and nonlinear optical properties of the studied compounds.