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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.
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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 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 $real^n$, $nge 1$: extendable, Jones, and almost continuous functions.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_infty(mu,X)$ and $L_1(mu,X)$.
The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space $C(X)$. In this paper, it is shown that for rather general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any linear bijection $T: A(X) to A(Y)$ such that $f geq 0$ if and only if $Tf geq 0$ gives rise to a homeomorphism $h: X to Y$ with which $T$ can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of uniformly continuous functions, Lipschitz functions and differentiable functions are presented.
Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism between $X$ and $Y$. This generalizes a classical result of Kaplansky concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order to extend the result further to noncompact spaces $X$ and $Y$. The main applications lie in the case when $X$ and $Y$ are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, local and bounded local
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