Schauder bases in locally solid lattice 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

In 2003, S. J. Dilworth, N. J. Kalton, D. Kutzarova and V. N. Temlyakov introduced the notion of almost greedy (respectively partially greedy) bases. These bases were characterized in terms of quasi-greediness and democracy (respectively conservativeness). In this paper, we show a new functional characterization of these type of bases in general Banach spaces following the spirit of the characterization of greediness proved in 2017 by P. M. Berná and Ó. Blasco.

This article closes the cycle of characterizations of greedy-like bases in the “isometric” case initiated in Albiac and Wojtaszczyk (J. Approx. Theory 138(1):65–86, 2006) with the characterization of 1-greedy bases and continued in Albiac and Ansorena (J. Approx. Theory 201:7–12, 2016) with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight estimate of the almost greedy constant of a basis in the non-isometric case.

Characterization of [formula omitted]-quasi-greedy bases

If E and F are locally convex lattices it is shown that the closure of the projective cone equals the biprojective cone for the projective topology on E 0 F and that this result cannot be extended to arbitrarily ordered locally convex spaces. The question of when these cones are b-cones is considered and applications are given to the tensor product of unconditional Schauder bases. In this note we investigate some properties of the projective and biprojec- tive cones in the tensor product of two locally convex lattices. In section 1 it is shown that the closure of the projective cone for the projective (n7) topology is the biprojective cone and an example is given to show that this result does not extend to arbitrary ordered locally convex spaces. The next section is concerned with the question of when the projective cone is a b-cone in the tensor product of ordered Banach spaces and we show that it is a b-cone for the projective topology but it is not, in general, a b-cone for the topology of bi-equicontinuous convergence. In the final section of the paper we use an order theoretic approach to study the tensor product of unconditional bases in Banach spaces. We show that the cone generated by the tensor product basis is just the biprojective cone and we give order criteria on when this basis is unconditional. Using these notions and the results of sections 1 and 2, we give examples of when the tensor product basis is unconditional and when it is conditional. Notation and terminology for this paper follows (12) and (15). 1. Cones in the Tensor Product. Let E and F be locally convex spaces

Banach spaces with a unique greedy basis

Now we reach the centerpiece of this volume, which is the theory of bases in Banach spaces. Since every Banach space is a vector space, it has a basis in the ordinary vector space sense, i.e., a set that spans and is linearly independent.However, this definition of basis restricts us to using only finite linear combinations of vectors, while in any normed space it makes sense to deal with infinite series. Restricting to finite linear combinations when working in an infinite-dimensional space is simply too restrictive for most purposes. Moreover, the proof that a vector space basis exists is nonconstructive in general, as it relies on the Axiom of Choice. Hence we need a new notion of basis that is appropriate for infinite-dimensional Banach spaces, and that is the main topic of this chapter.

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.

On uncountable unconditional bases in Banach spaces

00 that x = 2 n this basis is normalized so that \\yn = 1. In topological terms this means that (yn) is bounded and 'bounded away from zero' in the sense that there exists a neighbourhood V of 0 such that yn $ V for all n. These two properties are extremely useful in the theory of bases in Banach spaces. The purpose of this paper is to consider these properties for bases of general locally convex spaces.

Characterization of greedy bases in Banach spaces

Iwas a graduate student in mathematics at Princeton immediately following World War II, 1945-1947. In those days, with veterans of the war inundating universities, there was no option but to earn a Ph.D. within two years. My thesis research was in functional analysis, an area of abstract mathematics. My mentor was Professor S. Bochner, a distinguished contributor to harmonic analysis. My classmates included Richard Bellman (who later nurtured the method of dynamic programming), Kai Lai Chung (of Markov chain lore), Ted Harris (who developed broadly branching stochastic processes and their applications), Gil Hunt (who set the foundations of strong Markov processes, now commonly called Hunt processes), and Paul Meier (an applied statistician, known as the pioneer of the nonparametric Kaplan-Meier test). When I completed my degree in 1947, Professor S. Lefschetz, chairman of the Princeton mathematics department, selected me for the job of becoming the first Cal Tech Bateman Fellow. I was the youngest Ph.D. that year and did as I was told, although secretly I was happy to go to California. So, in September 1947, I got married and moved to Cal Tech. I labored that first year on mathematical problems of Banach spaces (e.g., bases in Banach spaces, normed rings), continuing my abstract mathematical education. During 1948, economic inflation was rampant in the country and I needed a second job. Henri Bohnenblust, chairman of the Cal Tech mathematics department, arranged a consulting job for me at the RAND Corporation. RAND at that time was sponsored by the Air Force and was engaged in matters of strategic planning for the cold war.

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