Projective limits of paratopological vector spaces (Q558384)

Various methods of constructing new paratopological spaces from given ones are described. First, a notion of a right dual of a paratopological vector space is introduced with the aim to define a right topology, namely, the right dual of a paratopological vector space \((X,\tau)\) is the space of functions \(X'_r=\{f:(X,\tau)\to(\mathbb R,u)\): \(f\) is linear and continuous\(\}\), where \(({\mathbb R},u)\) is the paratopological vector space introduced by the quasi-norm~\(u\) defined on~\({\mathbb R}\) by \(u(x)=\max \{x,0\}\). Then the author proves that in a certain class of normed spaces, the classical weak topology is determined by a right weak topology. Next, the quotient topology in the context of paratopological vector spaces is discussed. Finally, the projective limit of paratopological vector spaces is considered and it is proved that every pseudoconvex space is a projective limit of a quasi-normed space (here the quasi-norm \(q(\cdot)\) is a subadditive and positively homogeneous subnorm that satisfies the condition \(q(x)=q({-}x)=0 \iff x=0)\).