# Asymmetric Functional Analysis

## Abstract

An asymmetric seminorm is a positive sublinear functional *p* on a real vector space *X*. If *p*(*x*) = *p*(−*x*) = 0 implies *x* = 0, then *p* is called an asymmetric norm. The conjugate asymmetric seminorm is given by *p*(*x*) = *p*(−*x*), *x* ∈ *X*, and *ps* = *p* ∨ *p* is a seminorm, respectively a norm on *X* if *p* is an asymmetric norm. An important example is the asymmetric norm *u* on ℝ given by *u*(*t*) = *t*+, *t* ∈ ℝ, generating the upper topology on ℝ. In this case *u*(*t*) = *t*− and *u* ^{ s }(*t*) = ∣*t*∣, *t* ∈ ℝ. The dual *X* _{ p } of an asymmetric normed space (*X*, *p*) is formed by all upper semicontinuous linear functionals from (*X*, *p*) to (ℝ, ∣ ⋅ ∣), or equivalently, by all continuous linear functionals from (*X*, *p*) to (ℝ, *u*). In contrast to the usual case, *X* _{ p } is not a linear space but merely a cone contained in the dual *X*∗ of the normed space space (*X*, *ps*). The aim of this chapter is to present some basic results on asymmetric normed spaces, their duals and on continuous linear operators acting between them. Applications are given to best approximation in asymmetric normed spaces. As important examples one considers asymmetric norms on normed lattices and spaces of semi-Lipschitz functions on quasi-metric spaces. Asymmetric locally convex spaces are considered as well.

## Keywords

Normed Space Convex Body Cauchy Sequence Banach Lattice Weak Topology## Preview

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