It is proved the existence of large algebraic structures \break --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 non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.
We introduce the concept of {\em maximal lineability cardinal number}, $\mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $\HL(M)$, and lineability $\LL(M)$ of $M$. In particular, we will describe, in terms of $\LL$, the lineability and spaceability of the families of the following Darboux-like functions on $\real^n$, $n\ge 1$: extendable, Jones, and almost continuous functions.
Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Fa…
The area of research called \textquotedblleft Lineability\textquotedblright% \ looks for linear structures inside exotic subsets of vector spaces. In the last decade lineability/spaceability has been investigated in rather general settings; for instance, Set Theory, Probability Theory, Functional Analysis, Measure Theory, etc. It is a common feeling that positive results on lineability/spaceability are quite natural (i.e., in general \textquotedblleft large\textquotedblright subspaces can be found inside exotic subsets of vector spaces, in quite different settings) and more restrictive approac…
Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano's theorem on the existence of solutions of the ODE $u'=f(u)$ on $c_0$. The space of all continuous vector fields $f$ on $c_0$ is proved to contain a closed $\mathfrak{c}$-dimensional subspace formed by fields $f$ for which -- except for the null field -- the weak form of Peano's theorem fails to be true. The second application generalizes known results on the existence of closed $\mathfrak{c}$…
In 1931, Banach proved that, far from being exceptional objects, the Weierstrass functions form a residual set in the space $\mathcal{C}[0,1]$ of continuous functions. Later on, in 1966, V. I. Gurariy showed that, except for zero, there is an infinite-dimensional linear subspace of Weierstrass functions. This was the first example of \textit{lineability}. Over the last decade, this topic has attracted the continuous attention of the mathematical community, with a steady stream of papers being published, many of them in highly ranked mathematical journals. Several lineability criteria are known…
Given a continuous $n$-homogeneous polynomial $P\colon E\longrightarrow F$ between Banach spaces and $1\leq q\leq p<\infty$, in this paper we investigate some properties concerning lineability and spaceability of the $(p;q)$-summing set of $P$, defined by $S_{p;q}(P)=\{a\in E:P\mathrm{is}% (p;q)\mathrm{summing at}a\}$.
Let $m,n$ be positive integers. In this short note we prove that the set of all continuous and surjective functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ contains (excluding the 0 function) a $\mathfrak{c}$-dimensional vector space. This result is optimal in terms of dimension.
We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and $(α,β)$-lineability. Among other results, we prove that the class of multilinear operators taking values on $\ell_q \, (0<q \leq \infty)$ that are absolutely but not multiple summing is pointwise $\mathfrak{c}$-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also stud…
Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND.
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.
We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of $X_n$ contains $n$-dimensional subspaces, but no subspace of higher dimension. This gives a $n$-by-$n$ version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. We also study the existence of strongly proximinal subspaces of finite codimension, showing that for every $1<n<\infty$ and $1\leqslant k <n$, there is a Banach space…
Among other results we investigate $\left( α,β\right) $-lineability of the set of non-continuous $m$-linear operators defined between normed spaces as a subset of the space of all $m$-linear operators. We also give a partial answer to an open problem on the lineability of the set of non absolutely summing operators.
The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use \textit{ad hoc} arguments and few general techniques are known. Motivated by the search of general methods, in this paper we formally extend recent results of G.\ Botelho and V.V. Fávaro on invariant sequence spaces to a more general setting. Our main results show that some subsets of invariant sequence spaces contain, up to the null vector, a closed infinite-dimensional subspace.
The main result of the paper shows that, for 1<p and 1<=q, a linear operator T from l_p to l_q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p=1). For 1<p (and q different from p), as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T from l_p to l_q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p=q). We also investigate lineability of the sets of norm-attaining and non-norm attaini…
The Borel map takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. In the literature, it is well known that the restriction of this mapping to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In this paper, we are interested in studying how large the image of the Borel map is and we investigate the size and the structure of this image by using different approaches (Baire residuality, prevalence and lineability). We give an answer to this ques…
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinit…
We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda. \textit{Lineability, spaceability, and algebrability of certain subsets of function spaces,} Taiwanese J. Math., \textbf{13} (2009), no. 4, 1257--1269], and that it is strongly $\mathfrak{c}$-algebrable. We prove strong $\mathfrak{c}$-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of …
On strong spaceability of continuous functions and fractal dimensions
Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics（安徽财经大学统计与应用数学学院）；School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
In this work, we introduce some new generalized sequence space related to the space l(p). Furthermore we investigate some topological properties as the completeness, the isomorphism and also we give some inclusion relations between this sequence space and some of the other sequence spaces. In addition, we compute alpha-, beta- and gamma-duals of this space, and characterize certain matrix transformations on this sequence space.
In this paper, we investigate integrated and differentiated sequence spaces which emerge from the concept of the space bv of sequences of bounded variation.The integrated and differentiated sequence spaces which was initiated by Goes and Goes. The main propose of the present paper, we study of matrix domains and some properties of the integrated and differentiated sequence spaces. In section 3, we compute the alpha-, beta- and gamma duals of these spaces. Afterward, we characterize the matrix classes of these spaces with well-known sequence spaces.
In this paper we introduced the generalized difference Cesaro sequence spaces of fuzzy real numbers defined by Orlicz function. We study some topological properties of these spaces. We obtain some inclusion relations involving these sequence spaces. These notions generalize many notions on difference Cesaro sequence spaces.
We investigate algebraic structures within sets of surjective and injective linear operators between sequence spaces, completing results of Aron et al.
The new concepts are introduced of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we obtain several applications of this fact.
We analyze the construction of a sequence space $\widetildeΘ$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\widetildeΘ$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fréchet space $X_F=\cap_{s\in {\mathbb N}_0} X_s$, where $\{X_s\}_{s\in {\mathbb N}_0}$ is a sequence of Banach spaces.
We study the lattice of closed ideals in the algebra of continuous linear operators acting on $p$th Tandori and $p'$th Cesàro sequence spaces, $1\leqslant p<\infty$, which we show are isomorphic to the classical sequence spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and $(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.
We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.
The main purpose of this article is to introduce Pascal difference sequence spaces of fractional order $ τ$ over the sequence space $\ell_p$ and $\ell_\infty$. Some topological properties of these spaces are considered here along with the Schauder basis, $α -,β-$ and $γ-$duals of the spaces.
In this study, we define new paranormed sequence spaces by the sequences of Fibonacci numbers. Furthermore, we compute the $α-,β-$ and $γ-$ duals and obtain bases for these sequence spaces. Besides this, we characterize the matrix transformations from the new paranormed sequence spaces to the Maddox's spaces $c_{0}(q),c(q),\ell(q)$ and $\ell_{\infty}(q)$.
Here we have introduced the idea of rough Cauchyness of sequences in a cone metric space. Also here we have discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu.
The main purpose of the present paper is to introduce the space h_p and study of some properties of new sequence space. And we compute their dual spaces and characterizations of some matrix transformaitons.
In the present paper, by using the band matrix F defined by the Fibonacci sequence, we introduce the sequence sequence spaces c_0(F) and c(F). Also, we give some inclusion relations and construct the bases of the spaces c_0(F) and c(F). Finally, we compute the alpha-, beta-, gamma-duals of these spaces and characterize the classes (c_0(F),X) and (c(F),X) for certain choice of the sequence space X.
We analyze the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of $(E;p)$-summing multilinear mappings, a natural extension of the linear ideal of absolutely $(E;p)$-summing operators.
Imaninezhad and Miri introduced the sequence space $ d_{\infty} $ in order to characterize the continuous dual of the sequence space $ bv. $ We show by a counterexample that this claimed characterization is false.
We investigate convolution operators in the sequence spaces $d_p$, for $1\le p<\infty$. These spaces, for $p>1$, arise as dual spaces of the \ces sequence spaces $ces_p$ thoroughly investigated by G.~Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $\ell^p$.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that t…
We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.
We examine topological and algebraic genericity and spaceability for any pair $(X,Y)$, $X\subset Y$, $X\neq Y$ belonging to an extended chain of sequence spaces which contains the $\ell^p$ spaces, $0<p\leq \infty$.
We provide explicit commutative sequence space representations for classical function and distribution spaces on the real half-line. This is done by evaluating at the Fourier transforms of the elements of an orthonormal wavelet basis.
We obtain sequence space representations for a class of Fréchet spaces of entire functions with rapid decay on horizontal strips. In particular, we show that the projective Gelfand-Shilov spaces $Σ^1_ν$ and $Σ^ν_1$ are isomorphic to $Λ_{\infty}(n^{1/(ν+1)})$ for $ν> 0$.
From a given nuclear triplet we construct a nuclear triplet of sequence spaces and introduce a correlated Gaussian measure via the Bochner-Minlos theorem. Considering special types of correlation operators on such sequence spaces, certain conditional expectations can be given in an explicit way.
In this paper, we introduce some difference sequence spaces in bigeometric calculus. We determine the $α$-duals of these sequence spaces and study their matrix transformations. We also develop an interpolating polynomial in bigeometric calculus which is analogous to the classical Hermite interpolating polynomial.
