Basic Properties of Tempered ν-Sequence Spaces

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

The publication of Banach's book [2] in 1932 might be regarded as marking the beginning of the systematic study of Banach spaces. Research activity in this area has expanded dramatically during the past two decades. Interesting new directions have developed and interplays between Banach space theory and other mathematics have proved to be very valuable. Most well-known classical problems have been solved, but some remain and important new problems have arisen. Our purpose is very limited. It is to discuss some of the most important and fundamental facts about bases in Banach spaces, particularly those that may be interesting and useful for mathematicians in other fields. Often, only sketches of proofs will be given, while others are given in more detail because they are relatively easy and may contribute to developing intuitive feeling for the concepts involved. It will be seen that many very important and beautiful theorems have very easy and natural proofs. A basis (or Schauder basis) for a Banach space X is a sequence (en: n > 1) of members of X which has the property that, for each x in X, there is exactly one sequence of scalars (xi} for which x = xiei in the sense that lim.nX lix - E ixieijj = 0. Some important Banach spaces have very natural bases. If (en} is a complete orthonormal sequence in Hilbert space H and x is any member of H, then there is exactly one sequence (xn} of scalars such that lim x yn- lxieill = 0. For this sequence of scalars, each xi is (x, ei) and n 00 ~~~~~1/2

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

If a Banach space is saturated with subspaces with a Schauder basis, which embed into the linear span of any subsequence of their basis, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace.

Schauder bases in Banach spaces are studied in [5].In ordered Banach spaces a special type of Schauder bases, the O.P. Schauder bases, are studied because then the properties of ordered spaces can be used.

In this paper, we show that every Banach space with a Schauder basis can be seen as a totally ordered vector space. Indeed, this order can be considered as a lexicographical order since it is a generalization of lexicographical order in $\mathbb{R}^{n}.$ We also provide order structural properties of the order by approaching geometrical (cone) sense.

We study the existence of infima of subsets in Banach spaces ordered by normal cones associated to shrinking Schauder bases. Under these conditions we prove the existence of infima for a class of subsets verifying a weakly compactness property. Moreover we prove that a normal cone associated to a Schauder basis in a reflexive Banach space is strongly minihedral extending the known result for unconditional Schauder bases. Several examples are also discussed.

On Schauder bases in Lipschitz-free spaces

On Banach algebras of band-dominated operators and their order structure

Uncertain inverse problem for fractional dynamical systems using perturbed collage theorem

We prove the impossibility of expressing the space of entire functions on any infinite dimensional complex Banach space with a Schauder basis E as a countable union of spaces of entire functions that are bounded on countable open covers of E .

This article addresses the extension of classical spectral theory to the setting of non-separable Banach spaces (NSBS), a field that remains underdeveloped despite its increasing relevance in functional analysis and mathematical physics. While spectral theory in separable Banach spaces has been thoroughly investigated—particularly through the work of Fredholm, Riesz, and others—the transition to nonseparable spaces presents both technical and conceptual challenges. These arise mainly from the absence of countable dense subsets and orthonormal or Schauder bases, which are instrumental in formulating and proving classical theorems. The main objective of this work is to provide a rigorous and systematic framework for spectral theory in NSBS, focusing on the classification of spectral components, the generalisation of key theorems, and the topological and algebraic behaviour of operators acting on such spaces. We begin by recalling the foundations of spectral theory in the classical (separable) case and proceed to identify the structural modifications required in the non-separable context. A central part of the analysis is devoted to the residual and continuous spectra, whose properties exhibit significant deviations from their counterparts in separable settings. We propose new definitions, prove generalised versions of the spectral theorem, and characterise the structure of the spectrum under various topological assumptions. Particular attention is given to the stability of the spectrum under perturbations—both compact and non-compact—and to the conditions under which the classical spectral decomposition remains valid. Several illustrative examples are provided to highlight the influence of non-separability on spectral behaviour. The results presented here contribute to a deeper understanding of spectral phenomena in infinitedimensional analysis, particularly in cases where separability cannot be assumed. They offer theoretical tools for future work in operator theory, interpolation, partial differential equations, inverse problems, and the theory of operator algebras. This article thereby lays the groundwork for further mathematical developments in non-separable environments and suggests new directions for interdisciplinary applications.

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

A perfect sequence space X is said to be a step if /lc=A"=/M and A is a Banach space in its strong topology from Xx.In this paper a method is given to generate additional steps from a step X.Precisely, X" is a step where X"={x=(Xi)\xe C and \x\" = (\xi\")eX}, for 1</><co, with norm ||x!U=(|| xl'y1/.It is shown that X", l<p<x>, is reflexive iff X has a Schauder basis.The space of diagonal maps of X" into X is characterized, as is the space of diagonal nuclear maps of X into X" when X has a Schauder basis.

A set of simple flow domains\(\mathcal{D}\) relevant to turbulent flows and Schauder bases (Sect. 23.8 in Appendix A) for the phase space \(\Omega \) of functions defined on a domain \(\mathcal{D}\) of particular structure and its dual space \(\mathcal{N}\) called test function space are introduced for later use. A fundamental tool for operations in Banach and Hilbert spaces is the notion of a Schauder basis.