Total subspaces in dual Banach spaces which are not norming

The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping

Subspaces containing biorthogonal functionals of bases of different types

The paper is devoted to two particular cases of the following general problem. Let $\alpha$ and $\beta$ be two types of bases in Banach spaces. Let a Banach space $X$ has bases of both types and a subspace $M\subset X^*$ contains the sequence of biorthogonal functionals of some $\alpha$-basis in $X$. Does $M$ contain a sequence of biorthogonal functionals of some $\beta$-basis in $X$? The following particular cases are considered: $(\alpha, \beta)$=(Schauder bases, unconditional bases), $(\alpha, \beta)$=(Nonlinear operational bases, linear operational bases). The paper contains an investigation of some of the spaces constructed by S.Belle\-not in ``The $J$-sum of Banach spaces'', J. Funct. Anal. {\bf 48} (1982), 95--106. (These spaces are used in some examples.

Free Internal Waves in Polytropic Atmospheres

Free internal waves in polytropic atmospheres are studied (polytropic atmosphere is such one that the temperature of gas linearly depends on altitude). We suppose gas to be ideal and incompressible. Also, we regard the atmosphere of constant height with the "rigid lid" condition on its top to filter internal waves. If temperature, density and pressure of such undisturbed atmosphere do not depend on latitude and longitude then the internal waves are harmonic with apriori unknown eigenfrequencies, the problem permits separation of variables and reduces to the system of two ODE's. The first ODE (the Laplace's tidal equation) is analyzed by author earlier. The second ODE determines the vertical structure of the waves to be considered and has analytical solution for polytropic atmospheres. There are 6 dimensionless numbers, 2 for the Laplace's tidal equation and 4 for the vertical structure equation. The solution is a countable set of the eigenfrequencies and eigenfunctions of the vertical structure equation; every eigenfrequency/eigenfunction corresponds to its own countable set of the eigenfrequencies and eigenfunctions of the Laplace's tidal equation. Parametric analysis of the problem has been done. It shows that there exists the solution weakly depending on altitude-temperature variations and the atmosphere's height for parameters modelling the Earth's troposphere (with the "rigid lid" between the troposphere and the tropopause). The natural periods of internal waves have been obtained for this case.Comment: 7 pages, 1 table, 1 figur

Topologies on the set of all subspaces of a banach space and related questions of banach space geometry

For a Banach space $X$ we shall denote the set of all closed subspaces of $X$ by $G(X)$. In some kinds of problems it turned out to be useful to endow $G(X)$ with a topology. The main purpose of the present paper is to survey results on two the most common topologies on $G(X)$

Test-space characterizations of some classes of Banach spaces

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1) $X\notin\mathcal{P}$; (2) The spaces $\{T_\alpha\}_{\alpha\in A}$ admit uniformly bilipschitz embeddings into $X$. The first part of the paper is devoted to a simplification of the proof of the following test-space characterization obtained in M.I. Ostrovskii [Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., to appear]: For each sequence $\{X_m\}_{m=1}^\infty$ of finite-dimensional Banach spaces there is a sequence $\{H_n\}_{n=1}^\infty$ of finite connected unweighted graphs with maximum degree 3 such that the following conditions on a Banach space $Y$ are equivalent: (A) $Y$ admits uniformly isomorphic embeddings of $\{X_m\}_{m=1}^\infty$; (B) $Y$ admits uniformly bilipschitz embeddings of $\{H_n\}_{n=1}^\infty$. The second part of the paper is devoted to the case when $\{X_m\}_{m=1}^\infty$ is an increasing sequence of spaces. It is shown that in this case the class of spaces given by (A) can be characterized using one test-space, which can be chosen to be an infinite graph with maximum degree 3

A note on analytical representability of mappings inverse to integral operators

The condition onto pair ($F,G$) of function Banach spaces under which there exists a integral operator $T:F\to G$ with analytic kernel such that the inverse mapping $T^{-1}:$im$T\to F$ does not belong to arbitrary a priori given Borel (or Baire) class is found

On metric characterizations of the Radon-Nikod\'ym and related properties of Banach spaces

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond. The paper also contains related characterizations of reflexivity and the infinite tree property

Structure of total subspaces of dual Banach spaces

Let $X$ be a separable nonquasireflexive Banach space. Let $Y$ be a Banach space isomorphic to a subspace of $X^*$. The paper is devoted to the following questions: 1. Under what conditions does there exist an isomorphic embedding $T:Y\to X^*$ such that subspace $T(Y)\subset X^*$ is total? 2. If such embeddings exist, what are the possible orders of $T(Y)$? Here we need to recall some definitions. For a subset $M\subset X^*$ we denote the set of all limits of weak$^*$ convergent sequences in $M$ by $M_{(1)}$. Inductively, for ordinal number $\alpha$ we let $$M_{(\alpha)}=\cup_{\beta<\alpha}(M_{(\beta)})_{(1)}.$$ The least ordinal $\alpha$ for which $M_{(\alpha)}= M_{(\alpha+1)}$ is called the {\it order} of $M$

Steady 1D Stationary Currents of Spherical Gas Layer

Spherical layer of ideal gas is considered. The layer is in the sphere's gravity field. Existence possibility of steady 1D stationary currents of this layer is studied. This problem simulates zonal winds taking place in the atmospheres of some planets such as Venus, Titan, Jupiter and Saturn.Comment: 5 page

Radon-Nikod\'ym property and thick families of geodesics

Banach spaces without the Radon-Nikod\'ym property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family $T$ of geodesics joining points $u$ and $v$ in a metric space is called {\it thick} if there is $\alpha>0$ such that for every $g\in T$ and for any finite collection of points $r_1,...,r_n$ in the image of $g$, there is another $uv$-geodesic $\widetilde g\in T$ satisfying the conditions: $\widetilde g$ also passes through $r_1,...,r_n$, and, possibly, has some more common points with $g$. On the other hand, there is a finite collection of common points of $g$ and $\widetilde g$ which contains $r_1,...,r_n$ and is such that the sum of maximal deviations of the geodesics between these common points is at least $\alpha$