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algebraic balanced neighborhood Banach space barrelled bornivorous bornological bounded set bounded subset Cauchy filter Cauchy sequence closed hulls closed subspace coarser compact subset complete continuous linear form continuous linear map converges convex Hausdorff space convex neighborhood convex set corol Corollary defined definition denote direct sum dual system elements epimorphism equicontinuous subset example exerc exists a neighborhood F and G f is continuous filter basis finer finest locally convex finite family finite subset form a dual fundamental system Furthermore Hausdorff topology hence Hilbert space Hint homomorphism hyperplane infrabarrelled isomorphism lemma Let F locally convex Hausdorff locally convex space locally convex topology Mackey map f metric space Montel space open set precompact Proof Proposition prove Ptak space quotient space satisfies scalar Schwartz space semi-norms space and let space F subsets of F Suppose system of neighborhoods theorem topological space topological vector space topology induced topology on F zero