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Continuous operators on asymmetric normed spaces. (English) Zbl 1199.54165

For a real linear space, a function p:X + is called an asymmetric norm on X if for all x,yX and r + , (i) p(x)=p(-x)=0; (ii) p(rx)=rp(x); (iii) p(x+y)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,·), p(x)=x + with x + =sup{x,0} is an asymmetric norm on X.

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).

If (Y,q) is (,u) where u is the asymmetric norm on given by u(x)=x + , then LC(X,) and LC s (X,) are denoted by X * and X s* , respectively. It is proved that, with (X,·) and (Y,·) two normed lattices, p(x)=x + if xX and q(y)=y + if yY, fLC(X,Y) iff fLC s (X,Y) and f0; also p q * (f)f2p q * (f), for all fLC(X,Y) where p q * (f)=sup{q(f(x)):p(x)1}, it is further proved that, if (X,·) is a real normed lattice and q(x)=x + , then X s* =X * -X * .

In the last section, open mapping and closed graph theorems are given, in a suitable manner, in the new setting.

54E35Metric spaces, metrizability
54A05Topological spaces and generalizations
46A03General theory of locally convex spaces
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