The Mackey topology and coarser Mackey topologies The dual Mackey topology Some remarks on the Mackey topology in a Banach space A. J. Guirao1, V. Montesinos1 1Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Spain IX Encuentro de la Red de Análisis Funcional. Zafra, 2013 A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology Outline 1 The Mackey topology and coarser Mackey topologies 2 The dual Mackey topology A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. w(E; F) the weak topology on E associated to hE; Fi. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology associated to a dual pair hE; Fi a dual pair. w(E; F) the weak topology on E associated to hE; Fi. w(E; F) is the coarsest lcs topology on E compatible with hE; Fi. K all the absolutely convex w(F; E)-compact subsets of F. µ(E; F) := TK, the Mackey topology on E. Theorem (Mackey–Arens) µ(E; F) is the finest lcs topology on E compatible with hE; Fi. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology Grothendieck’s construction of the completion Theorem (Grothendieck) hE; Fi dual pair. M saturated family of bounded subsets of F that covers F. Then E\[TM] =all linear functionals f : F ! R st f jM w(F; E)-continuous 8M 2 M. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology in a Banach space and its dual A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (Eberlein) (E; T ) µ(E; E∗)-(quasi)complete. Then w-RNK ) w-RK. Theorem (Krein) (E; T ) µ(E; E∗)-(quasi)complete, K w-compact. Then Γ(K ) w-compact. Theorem (James) (E; T ) µ(E; E∗)-(quasi)complete, all f 2 E∗ attains the sup on a w-closed K . Then K is w-compact. The Mackey topology and coarser Mackey topologies The dual Mackey topology The role of the Mackey topology in a lcs (Eberlein, Krein, and James theorems) A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (Krein) (E; T ) µ(E; E∗)-(quasi)complete, K w-compact. Then Γ(K ) w-compact. Theorem (James) (E; T ) µ(E; E∗)-(quasi)complete, all f 2 E∗ attains the sup on a w-closed K . Then K is w-compact. The Mackey topology and coarser Mackey topologies The dual Mackey topology The role of the Mackey topology in a lcs (Eberlein, Krein, and James theorems) Theorem (Eberlein) (E; T ) µ(E; E∗)-(quasi)complete. Then w-RNK ) w-RK. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (James) (E; T ) µ(E; E∗)-(quasi)complete, all f 2 E∗ attains the sup on a w-closed K . Then K is w-compact. The Mackey topology and coarser Mackey topologies The dual Mackey topology The role of the Mackey topology in a lcs (Eberlein, Krein, and James theorems) Theorem (Eberlein) (E; T ) µ(E; E∗)-(quasi)complete. Then w-RNK ) w-RK. Theorem (Krein) (E; T ) µ(E; E∗)-(quasi)complete, K w-compact. Then Γ(K ) w-compact. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology The role of the Mackey topology in a lcs (Eberlein, Krein, and James theorems) Theorem (Eberlein) (E; T ) µ(E; E∗)-(quasi)complete. Then w-RNK ) w-RK. Theorem (Krein) (E; T ) µ(E; E∗)-(quasi)complete, K w-compact. Then Γ(K ) w-compact. Theorem (James) (E; T ) µ(E; E∗)-(quasi)complete, all f 2 E∗ attains the sup on a w-closed K . Then K is w-compact. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (Howard, 1973) X Banach. Then µ(X ∗; X)-RSK ) µ(X ∗; X)-RK. Theorem X Banach. Then µ(X ∗; X)-RNK ) µ(X ∗; X)-RK. The Mackey topology and coarser Mackey topologies The dual Mackey topology A theorem of Howard (and an extension) (R)SK)(R)NK((R)K A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem X Banach. Then µ(X ∗; X)-RNK ) µ(X ∗; X)-RK. The Mackey topology and coarser Mackey topologies The dual Mackey topology A theorem of Howard (and an extension) (R)SK)(R)NK((R)K Theorem (Howard, 1973) X Banach. Then µ(X ∗; X)-RSK ) µ(X ∗; X)-RK. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology A theorem of Howard (and an extension) (R)SK)(R)NK((R)K Theorem (Howard, 1973) X Banach. Then µ(X ∗; X)-RSK ) µ(X ∗; X)-RK. Theorem X Banach. Then µ(X ∗; X)-RNK ) µ(X ∗; X)-RK. A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space The Mackey topology and coarser Mackey topologies The dual Mackey topology In general, µ(X ∗; X) is not angelic A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Question (rephrased): X Banach. Y ⊂ X ∗ k · k-closed, w ∗-dense. Is (X; µ(X; Y )) always complete? Theorem (Bonet–Cascales, 2010) (`1[0; 1]; µ(`1[0; 1]; C[0; 1])) is not complete. Variation of an example of Sánchez and Suárez-Granero, 2007. The Mackey topology and coarser Mackey topologies The dual Mackey topology Completeness for Mackey topologies coarser than k · k (a question of Kunze and Arendt, 2008) A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem (Bonet–Cascales, 2010) (`1[0; 1]; µ(`1[0; 1]; C[0; 1])) is not complete. Variation of an example of Sánchez and Suárez-Granero, 2007. The Mackey topology and coarser Mackey topologies The dual Mackey topology Completeness for Mackey topologies coarser than k · k (a question of Kunze and Arendt, 2008) Question (rephrased): X Banach. Y ⊂ X ∗ k · k-closed, w ∗-dense. Is (X; µ(X; Y )) always complete? A. J.