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Zbl 1199.54165
Alegre, C.
Continuous operators on asymmetric normed spaces. (English)
[J] Acta Math. Hung. 122, No. 4, 357-372 (2009). ISSN 0236-5294; ISSN 1588-2632/e

For a real linear space, a function $p:X\to \Bbb R^+$ is called an asymmetric norm on $X$ if for all $x,y\in X$ and $r\in \Bbb R^+$, (i) $p(x)=p(-x)=0$; (ii) $p(rx)=rp(x)$; (iii) $p(x+y)\le p(x)+p(y)$. For an asymmetric norm $p$ on $X$, $p^{-1}$, defined on $X$ by $p^{-1}(x)=p(-x)$ is also an asymmetric norm on $X$; the function $p^s$ defined on $X$ by $p^s(x)=\max\{p(x),p^{-1}(x)\}$ is obviously a norm on $X$; also, for a normed lattice $(X,\Vert \,.\,\Vert )$, $p(x)=\Vert x^+\Vert $ with $x^+=\sup\{x,0\}$ is an asymmetric norm on $X$. \par The author uses the symbol $LC(X,Y)$ to denote the set of all continuous linear mappings from $(X,p)$ to $(Y,q)$ where $p$, $q$ are asymmetric norms whereas $LC^s(X,Y)$ is used to denote the set of all continuous linear mappings from $(X,p^s)$ to $(Y,q^s)$; $LC(X,Y)$ is not a linear space but a cone which is included in $LC^s(X,Y)$. \par If $(Y,q)$ is $(\Bbb R,u)$ where $u$ is the asymmetric norm on $\Bbb R$ given by $u(x)=x^+$, then $LC(X,\Bbb R)$ and $LC^s(X,\Bbb R)$ are denoted by $X^*$ and $X^{s*}$, respectively. It is proved that, with $(X,\Vert \, .\,\Vert )$ and $(Y,\Vert \, .\,\Vert )$ two normed lattices, $p(x)=\Vert x^+\Vert $ if $x\in X$ and $q(y)=\Vert y^+\Vert $ if $y\in Y$, $f\in LC(X,Y)$ iff $f\in LC^s(X,Y)$ and $f\ge 0$; also $p^*_q(f)\le \Vert f\Vert \le 2p^*_q(f)$, for all $f\in LC(X,Y)$ where $p^*_q(f)=\sup\{q(f(x)):p(x)\le 1\}$, it is further proved that, if $(X,\Vert \, .\,\Vert )$ is a real normed lattice and $q(x)=\Vert x^+\Vert $, then $X^{s*}=X^*-X^*$. \par In the last section, open mapping and closed graph theorems are given, in a suitable manner, in the new setting.
[M. N. Mukherjee (Calcutta)]

MSC 2000:: *54E35 Metric spaces, metrizability
54A05 Topological spaces and generalizations
46A03 General theory of locally convex spaces

Keywords: asymmetric norm; asymmetric normed linear space; cone; quasi-metric; semicontinuous linear map

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