Topological Vector Spaces

absolutely convex envelope absolutely convex neighbourhood absorbent algebraic applied Banach space barrelled barrelled space base of neighbourhoods bilinear bounded sets bounded subsets bourhood called Cauchy Chapter clearly closed absolutely convex closed graph closed vector subspace closure compact set contained continuous linear mapping convergence convex topology COROLLARY corresponding defined definition denoted direct sum dual pair elements equicontinuous equivalent example fact filter finer finite follows Fréchet space fully complete functions give given Hence identity implies inductive limit intersection isomorphism Lemma meeting metrisable nearly normed space origin polar precompact projective limit Proof Prop PROPOSITION prove quotient result satisfied scalar seminorms separated convex space sequence space E space F spanned strand strong Suppl supplement Suppose taking theorem topology of uniform uniform convergence union vector space vector subspace weakly