Locally Convex Spaces (Graduate Texts in Mathematics, Volume 269)

Evidence. consider X is an infrabarreled Hausdorff in the neighborhood convex house. If is weak-∗ compact in X ∗, then as a subset of X ∗ with the weak-∗ topology: D is nonempty, bounded, closed, convex, balanced, and entire (Corollary 1. 32), for this reason D is sequentially entire. If A is bounded in X, then A ∘ is a barrel in X ∗ which needs to now take up D through the absorption precept. accordingly D ∘ absorbs A [Theorem three. 20(e)]. when you consider that A is bigoted and X is infrabarreled, D ∘ is an unique local of zero. seeing that D was once arbitrary, the Mackey topology can't be strictly finer than the unique topology. Now for “bornological,” an issue that might reappear in Sect. four. three, then mostly disappear. The bornological doesn't require completeness; faraway from it. Proposition four. 10. consider X is a Hausdorff in the neighborhood convex area. If X is first countable, then X is bornological. facts. believe X is a primary countable Hausdorff in the neighborhood convex house. pick out a countable local base V 1, V 2, … at zero such that (Theorem 1. thirteen offers extra, yet this can be all we'd like. ) believe B is a convex, balanced subset of X. instead of assuming that B absorbs all bounded units after which proving that B is an area of zero, we will think that B isn't an area of zero and build a bounded (in truth, compact) set that B doesn't take in. imagine B isn't really an area of zero. Then for all n, , so . opt for x n ∈ V n − nB. because x n ∈ V n and : x n → zero, so is compact, as a result bounded. yet x n ∉ nB says that B can't take up . Corollary four. eleven. Normed areas, Fréchet areas, and LF-spaces are bornological. evidence. Normed areas and Fréchet areas are first countable. As for LF-spaces, believe that's an LF-space, and B is a balanced, convex subset of X that absorbs all bounded units. If A is bounded in X n , then every one non-stop linear practical on X restricts to a continuing linear sensible on X n (Proposition three. 39) and so is bounded on A. for that reason A is bounded in X via Corollary three. 31, so B absorbs A. that's, B ∩ X n is a convex balanced subset of X n that absorbs A. due to the fact A was once arbitrary and X n is bornological, B ∩ X n is a local of zero in X n . due to the fact that n is bigoted, B is, by way of definition, a member of the unique local base at zero defining the LF-topology of X. The previous facts for LF-spaces works as the LF-topology base was once outlined with out assuming “ is open in X n ” in Sect. three. eight. Now for the “point” of assuming the bornological . Theorem four. 12. feel X and Y are Hausdorff in the community convex areas, and T : X → Y is a linear transformation. reflect on the next 3 statements: (i) T is constant. (ii) If x n → zero in X, then T(x n ) → zero in Y. (iii) T is bounded, that's T(A) is bounded in Y every time A is bounded in X. Then (i) ⇒ (ii) ⇒ (iii) continually, and (iii) ⇒ (i) if X is bornological. evidence. (i) ⇒ (ii), due to the fact continuity ⇒ sequential continuity. imagine (ii): If A is bounded yet T(A) isn't really bounded in Y, decide on a local V of zero in Y that doesn't take up T(A). opt for T(x n ) ∈ T(A) − nV, x n ∈ A. Then through Proposition 2.