Juan B. Seoane-Sepúlveda

Linear subsets of nonlinear sets in topological vector spaces

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars

The Bohnenblust-Hille inequality was obtained in 1931 and ( in the case of real scalars) asserts that for every positive integer m there is a constant Cm so that ((N)Sigma(i1 , . . . , im=1)vertical bar T(e(i1) (,...,) e(im))vertical bar(2m/m+1))(m+1/2) <= C-m parallel to T parallel to for all positive integers N and every m-linear mapping T : l(infinity)(N) x...x l(infinity)(N) -> R. Since then, several authors have obtained upper estimates for the values of C-m. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for C-m.

Some results and open questions on spaceability in function spaces

A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y subset of M boolean OR {0}. In this article we prove that, for every infinite dimensional closed subspace X of C[0, 1], the set of functions in X having infinitely many zeros in [0, 1] is spaceable in X. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0, 1] or Muntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0, 1], as well as oscillating and annulling properties of subspaces of C[0, 1].

Sierpinski-Zygmund functions and other problems on lineability

We find large algebraic structures inside the following sets of pathological functions: (i) perfectly everywhere surjective functions, (ii) differentiable functions with almost nowhere continuous derivatives, (iii) differentiable nowhere monotone functions, and (iv) Sierpinski-Zygmund functions. The conclusions obtained on (i) and (iii) are improvements of some already known results.

A Characterization of Continuity Revisited

Abstract It is well known that a function f : ℝ → ℝ is continuous if and only if the image of every compact set under f is compact and the image of every connected set is connected. We show that there exist two 2C-dimensional linear spaces of nowhere continuous functions that (except for the zero function) transform compact sets into compact sets and connected sets into connected sets respectively.

Estimates for the asymptotic behaviour of the constants in the Bohnenblust–Hille inequality

A classical inequality due to H.F. Bohnenblust and E. Hille states that for every positive integer n there is a constant C n  > 0 so that for every positive integer N and every n-linear mapping . The original estimates for those constants from Bohnenblust and Hille are In this note we present explicit formulae for quite better constants, and calculate the asymptotic behaviour of these estimates, completing recent results of the second and third authors. For example, we show that, if C ℝ,n and C ℂ,n denote (respectively) these estimates for the real and complex Bohnenblust–Hille inequality then, for every even positive integer n, for a certain sequence {r n } which we estimate numerically to belong to the interval (1, 3/2) (the case n odd is similar). Simultaneously, assuming that {r n } is in fact convergent, we also conclude that

CHAOS ON FUNCTION SPACES

We give a sufficient condition for an operator to be chaotic and we use this condition to show that, in the Banach space C-0[0, infinity) the operator (T(lambda,c)f) (t) = lambda f (t + c) (with lambda > 1 and c > 0) is chaotic, with every n is an element of N being a period for this operator. We also describe a technique to construct, explicitly, hypercyclic functions for this operator.

When the Identity Theorem "Seems" to Fail

Abstract The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A ⊂C∞ (ℝ); (iv) every element of A has infinitely many zeros in ℝ; and (v) for every f ε A 0 and n ε N, f(n) (the nth derivative of f) enjoys the same properties as the elements in A0 . This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.

Lineability in sequence and function spaces

We prove the existence of large algebraic structures - including large vector subspaces or infinitely generated free algebras - inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of discontinuous separately continuous real functions. Lineability in special spaces of sequences is also investigated.Programa de Doutorado Sanduiche no Exterior (Coordenacao de aperfeicoamento de pessoal de nivel superior)

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in is exactly . We also give the exact value of the real polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in . Finally, we provide lower estimates for the real polynomial Bohnenblust–Hille constant for polynomials in of higher degrees.