Topological Vector Spaces

absolutely convex envelope absolutely convex neighbourhood absorbent algebraic applied Banach space barrelled base of neighbourhoods bilinear bounded sets bounded subsets called Cauchy Chapter clearly closed absolutely convex closed graph closed vector subspace closure compact set contained continuous linear mapping convergence convex topology COROLLARY defined definition denoted direct sum dual pair elements equicontinuous equivalent example fact filter finer finest finite follows Fréchet space fully complete functions give given Hence identity implies inductive limit intersection isomorphism Lemma meet 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 theorem topology of uniform uniform convergence union vector space vector subspace weakly