Continuous operators on asymmetric normed spaces

Abstract

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y ) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = x + , q(y) = y + ), we prove that the positive operators from X to Y are elements of the cone LC(X, Y ).We also study the dual space of an asymmetric normed space and nally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases.

Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

“…The following version of the open mapping theorem was proved by Alegre [1]. and show that ϕ isq-σ-subadditive.…”

The weak topology in finite dimensional asymmetric normed spaces

“…Let {e i } n i=1 the canonical basis of R n and let {e * i } n i=1 its dual basis, i.e., e * i (e j ) = δ i,j . Since e * i is continuous in (R n , • ) and e * i ≥ 0, by Corollary 3 of [1], we have that…”

“…The set X * is called the dual space of (X, q). More information about these spaces can be found in [1], [2] and [7].…”

“…An important class of asymmetric normed spaces are those where the asymmetric norm is given by a lattice norm. If (X, • , ≤) is a normed lattice, the function q defined on X by q(x) = x + , with x + = x ∨ 0, is an asymmetric norm on X and the norm q s is equivalent to the norm • (see, for example, [1,5] ). We shall refer to q as an asymmetric lattice norm on X.…”

A Characterization of Completeness via Absolutely Convergent Series and the Weierstrass Test in Asymmetric Normed Semilinear Spaces

“…On account of [3], a systematic and deep study of asymmetric normed linear spaces and other related structures such as asymmetric normed semilinear spaces has been made by Romaguera and some of his collaborators. Many of the aforesaid results can be found in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”