Highly tempering infinite matrices

[EN] In this short note, it is proved the existence of infinite matrices that not only preserve convergence and limits of sequences but also convert every member of some dense vector space consisting, except for zero, of divergent sequences, into a convergent sequence.The first author has been supported by the Plan Andaluz de Investigacion de la Junta de Andalucia FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The second and third authors have been supported by MEC, Grant MTM2016-75963-P. The fourth author has been supported by Grant MTM2015-65825-P.Bernal-González, L.; Conejero, JA.; Murillo Arcila, M.; Seoane-Sepúlveda, J. (2018). Highly tempering infinite matrices. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 112(2):341-345. https://doi.org/10.1007/s13398-017-0385-8S3413451122Aizpuru, A., Pérez-Eslava, C., Seoane-Sepúlveda, J.B.: Linear structure of sets of divergent sequences and series. Linear Algebra Appl. 418(2–3), 595–598 (2006)Aron, R.M., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2016)Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on $${\mathbb{R}}$$ R . Proc. Am. Math. Soc. 133(3), 795–803 (2005)Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014). doi: 10.1090/S0273-0979-2013-01421-6Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in $${\mathbb{R}}^{\mathbb{R}}$$ R R . J. Math. Anal. Appl. 401(2), 959–962 (2013)Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. Dover Publications Inc., Mineola, NY (2003)Horváth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading, MA (1966)Peyerimhoff, A.: Lectures on Summability. Lecture Notes in Mathematics. Springer, Berlin (1970)Kuratowski, K., Mostowski, A.: Set Theory. North Holland, Amsterdam (1976)Wilansky, A.: Summability Through Functional Analysis. North-Holland, Amsterdam (1984

L-p[0,1] \ boolean OR(q > p) L-q[0,1] is spaceable for every p > 0

In this short note we prove the result stated in the title: that is, for every p > 0 there exists an infinite dimensional closed linear sub-space of L-p[0, 1] every nonzero element of which does not belong to boolean OR(q>p) L-q[0, 1]. This answers in the positive a question raised in 2010 by R.M. Aron on the spaceability of the above sets (for both, the Banach and quasi-Banach cases). We also complete some recent results from Botelho et al. (2011) [3] for subsets of sequence spaces. (C) 2012 Elsevier Inc. All rights reserved

On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality

The search for sharp constants for inequalities of the type Littlewood's 4/3 and Bohnenblust-Hille, besides its pure mathematical interest, has shown unexpected applications in many different fields, such as Analytic Number Theory, Quantum Information Theory, or (for instance) in deep results on the $n$-dimensional Bohr radius. The recent estimates obtained for the multilinear Bohnenblust-Hille inequality (in the case of real scalars) have been recently used, as a crucial step, by A. Montanaro in order to solve problems in the theory of quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust-Hille constants in the case of complex scalars. For bilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3 inequality (in the case of real scalars) when the exponent 4/3 is replaced by any $r\geq4/3.$ As a consequence of our estimates we show that the optimal constants for the real case are always strictly greater than the constants for the complex case

There exist multilinear Bohnenblust-Hille constants (Cn)n=1(infinity) with limn ->infinity(Cn+1-Cn)=0

The n-linear Bohnenblust-Hille inequality asserts that there is a constant C-n is an element of [1, infinity) such that the l(2n/n+1)-norm of (U(e(i1), ..., e(in)))(i1, ...,in=1)(N) is bounded above by C-n times the supremum norm of U, for any n-linear form U :C-N x ... x C-N -> C and N is an element of N (the same holds for real scalars). We prove what we call Fundamental Lemma, which brings new information on the optimal constants, (K-n)(n=1)(infinity) for both real and complex scalars. For instance, Kn+1 - K-n = 2. We study the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities and provide estimates for Bohnenblust-Hille-type inequality constants for any exponent q is an element of [2n/n+1, infinity). (C) 2012 Elsevier Inc. All rights reserved

Lineability and algebrability of pathological phenomena in analysis

We show that, in analysis, many pathological phenomena occur more often than one could expect, that is, in a linear or algebraic way. We show this by means of the construction of large algebraic structures (infinite dimensional vector spaces or infinitely generated algebras) enjoying some special or pathological properties

Linear structure of sets of divergent sequences and series

We show that there exist infinite dimensional spaces of series, every non-zero element of which, enjoys certain pathological property. Some of these properties consist on being (i) conditional convergent, (ii) divergent, or (iii) being a subspace of l(infinity) of divergent series. We also show that the space 1(1)(omega)(X) of all weakly unconditionally Cauchy series in X has an infinite dimensional vector space of non-weakly convergent series, and that the set of unconditionally convergent series on X contains a vector space E, of infinite dimension, so that if x is an element of E \ {0} then Sigma(i) parallel to x(i)parallel to = infinity

The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal

The search of sharp estimates for the constants in the Bohnenblust-Hille inequality, besides its challenging nature, has quite important applications in different fields of mathematics and physics. For homogeneous polynomials, it was recently shown that the Bohnenblust-Hille inequality (for complex scalars) is hypercontractive. This result, interesting by itself, has found direct striking applications in the solution of several important problems. For multilinear mappings, precise information on the asymptotic behavior of the constants of the Bohnenblust-Hille inequality is of particular importance for applications in Quantum Information Theory and multipartite Bell inequalities. In this paper, using elementary tools, we prove a quite surprising result: the asymptotic growth of the constants in the multilinear Bohnenblust-Hille inequality is optimal. Besides its intrinsic mathematical interest and potential applications to different areas, the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years (such as, for instance, the multilinear version of the famous Grothendieck theorem for absolutely summing operators) always present constants C-m's growing at an exponential rate of certain power of m