On bases in Banach spaces

ACTA UNIVERSITATIS APULENSIS, 2015

It is well known that every vector space has a Hamel basis. In this short note we give a novel proof to show that for an infinite-dimensional Hilbert space, a basis is never a Hamel basis. 2010 Mathematics Subject Classification: 46C05.

British Journal of Mathematics & Computer Science, 2014

We introduce the notion of summable bases that naturally generalizes the notion of unconditional sequence bases for Banach spaces. We shall be particularly interested in some classical results on sequences and series in separable Banach spaces that carry over or naturally extend to the case of non-separable Banach spaces.

Studia Mathematica, 2008

We show that an infinite-dimensional complete linear space X has: • a dense hereditarily Baire Hamel basis if |X| ≤ c + ; • a dense non-meager Hamel basis if |X| = κ ω = 2 κ for some cardinal κ.

Israel Journal of Mathematics, 1996

Let X be a Banach space with an unconditional finite-dimensional Schauder decomposition (E n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis for X by forming an unconditional basis for each E n. For example, we show that if sup dim E n < ∞ and X has Gordon-Lewis local unconditional structure then X has an unconditional basis of this type. We also give an example of a non-Hilbertian space X with the property that whenever Y is a closed subspace of X with a UFDD (E n) such that sup dim E n < ∞ then Y has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space K of systems of scalars are considered in this work. The concepts of uncountable K-Bessel, K-Hilbert systems, K-frames and K *-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable K-Besselianness, K-Hilbertianness for systems, K-frames and unconditional K *-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of K-Besselian systems which are not minimal are given. It is proved that every K-Hilbertian systems is minimal. The case where K is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.

Springer Tracts in Natural Philosophy, 1969

Acta Mathematica Sinica, 2014

Mathematical Notes of the Academy of Sciences of the USSR, 1988

Complex Analysis and Operator Theory, 2018

The construction of bases comprising of elements with special structures in various Banach and Hilbert spaces is a major enterprise in many applications. The current article is targeted to study what we call perturbed bases and frames for Banach and Hilbert spaces. To this end, we consider a pair of bounded linear operators related via an inequality of a special form. Our results are in spirit close to a family of bounded linear operators known in fractal approximation theory. Hence, the proposed theory exhibits, in particular, bases and frames consisting of fractal (self-referential) functions for some standard spaces of functions. In the last part of the article we study the structure of the space of (strictly) relatively bounded operators. Keywords Schauder basis • Topological isomorphism • Fractal operator • Space of relatively bounded operators Mathematics Subject Classification 46B15 • 47B38 • 47L05 • 28A80 Communicated by Daniel Aron Alpay.

By using some generalized Riemann integrals instead of ordinary sums and multiplication systems of Banach spaces instead of Banach spaces, we establish some natural generalizations of the most basic facts on Schauder bases so that Hamel bases, and some other important unconditional bases, could also be included.