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 |M w(F, E)-continuous ∀M ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space 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? Theorem (Bonet–Cascales, 2010)

(`1[0, 1], µ(`1[0, 1], C[0, 1])) is not complete.

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 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? 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.

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 Completeness for Mackey topologies coarser than k · k (a question of Kunze and Arendt, 2008)

Theorem (Bonet–Cascales, 2010)

(`1[0, 1], µ(`1[0, 1], C[0, 1]) is not complete.

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 case of separable Banach spaces

Theorem X Banach separable. P ⊂ X ∗ norming, k · k-closed, (X, µ(X, P)) complete. P ⊂ Y ⊂ X ∗,Y k · k-closed. Then (X, µ(X, Y )) is complete.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space w ∗ S := {all x∗ ∈ X ∗, ∃K ∈ K, K ⊂ P ⊕ [x∗], x∗ ∈ K ∩ P }.

Theorem k·k (i) If µ(X, P) complete, S0 ⊂ S,Y = P ⊕ span{S0} , then µ(X, Y ) is complete. (ii) x∗ 6∈ S, then (X, µ(X, P ⊕ [x∗])) not complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Another approach to Bonet–Cascales solution

X Banach. K all absolutely convex w ∗-compact subsets of X ∗, P ⊂ X ∗, w ∗-dense, k · k-closed.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem k·k (i) If µ(X, P) complete, S0 ⊂ S,Y = P ⊕ span{S0} , then µ(X, Y ) is complete. (ii) x∗ 6∈ S, then (X, µ(X, P ⊕ [x∗])) not complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Another approach to Bonet–Cascales solution

X Banach. K all absolutely convex w ∗-compact subsets of X ∗, P ⊂ X ∗, w ∗-dense, k · k-closed.

w ∗ S := {all x∗ ∈ X ∗, ∃K ∈ K, K ⊂ P ⊕ [x∗], x∗ ∈ K ∩ P }.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space (ii) x∗ 6∈ S, then (X, µ(X, P ⊕ [x∗])) not complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Another approach to Bonet–Cascales solution

X Banach. K all absolutely convex w ∗-compact subsets of X ∗, P ⊂ X ∗, w ∗-dense, k · k-closed.

w ∗ S := {all x∗ ∈ X ∗, ∃K ∈ K, K ⊂ P ⊕ [x∗], x∗ ∈ K ∩ P }.

Theorem k·k (i) If µ(X, P) complete, S0 ⊂ S,Y = P ⊕ span{S0} , then µ(X, Y ) is complete.

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 Another approach to Bonet–Cascales solution

X Banach. K all absolutely convex w ∗-compact subsets of X ∗, P ⊂ X ∗, w ∗-dense, k · k-closed.

w ∗ S := {all x∗ ∈ X ∗, ∃K ∈ K, K ⊂ P ⊕ [x∗], x∗ ∈ K ∩ P }.

Theorem k·k (i) If µ(X, P) complete, S0 ⊂ S,Y = P ⊕ span{S0} , then µ(X, Y ) is complete. (ii) x∗ 6∈ S, then (X, µ(X, P ⊕ [x∗])) not complete.

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 Another approach to Bonet–Cascales solution

Theorem k·k (i) If µ(X, P) complete, S0 ⊂ S,Y = P ⊕ span{S0} , then µ(X, Y ) is complete. (ii) x∗ 6∈ S, then (X, µ(X, P ⊕ [x∗])) not complete.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space ∗ (ii) x ∈ `∞(Γ), supp (x ) uncountable. Then ∗ (`1(Γ), µ(`1(Γ), c0(Γ) ⊕ [x ])) non-complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Another approach to Bonet–Cascales solution

Γ uncountable. Corollary ∗ ∗ (i) x ∈ `∞(Γ), supp (x ) countable. Then ∗ (`1(Γ), µ(`1(Γ), c0(Γ) ⊕ [x ])) is complete.

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 Another approach to Bonet–Cascales solution

Γ uncountable. Corollary ∗ ∗ (i) x ∈ `∞(Γ), supp (x ) countable. Then ∗ (`1(Γ), µ(`1(Γ), c0(Γ) ⊕ [x ])) is complete. ∗ (ii) x ∈ `∞(Γ), supp (x ) uncountable. Then ∗ (`1(Γ), µ(`1(Γ), c0(Γ) ⊕ [x ])) non-complete.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space ∗∗ ∗∗ x ∈ `∞ \ c0, H := ker x . If (H, w(H, c0)) locally complete ⇒ (c0, µ(c0, H)) complete. `1 6⊂ X separable nonreflexive. Then (X, µ(X, H)) locally complete ⇒ (H, µ(H, X)) complete. Theorem ∗ ∀ H ⊂ c0, k · k1-closed norming. Then (c0, w(c0, H)) locally complete, hence (c0, µ(c0, H)) is complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Spaces without a predual

X Banach. H ⊂ X ∗ closed norming subspace iff H = kerx∗∗, x∗∗ ∈ X ∗∗ \ X.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space `1 6⊂ X separable nonreflexive. Then (X, µ(X, H)) locally complete ⇒ (H, µ(H, X)) complete. Theorem ∗ ∀ H ⊂ c0, k · k1-closed norming. Then (c0, w(c0, H)) locally complete, hence (c0, µ(c0, H)) is complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Spaces without a predual

X Banach. H ⊂ X ∗ closed norming subspace iff H = kerx∗∗, x∗∗ ∈ X ∗∗ \ X. ∗∗ ∗∗ x ∈ `∞ \ c0, H := ker x . If (H, w(H, c0)) locally complete ⇒ (c0, µ(c0, H)) complete.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem ∗ ∀ H ⊂ c0, k · k1-closed norming. Then (c0, w(c0, H)) locally complete, hence (c0, µ(c0, H)) is complete.

The Mackey topology and coarser Mackey topologies The dual Mackey topology Spaces without a predual

X Banach. H ⊂ X ∗ closed norming subspace iff H = kerx∗∗, x∗∗ ∈ X ∗∗ \ X. ∗∗ ∗∗ x ∈ `∞ \ c0, H := ker x . If (H, w(H, c0)) locally complete ⇒ (c0, µ(c0, H)) complete. `1 6⊂ X separable nonreflexive. Then (X, µ(X, H)) locally complete ⇒ (H, µ(H, X)) complete.

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 Spaces without a predual

X Banach. H ⊂ X ∗ closed norming subspace iff H = kerx∗∗, x∗∗ ∈ X ∗∗ \ X. ∗∗ ∗∗ x ∈ `∞ \ c0, H := ker x . If (H, w(H, c0)) locally complete ⇒ (c0, µ(c0, H)) complete. `1 6⊂ X separable nonreflexive. Then (X, µ(X, H)) locally complete ⇒ (H, µ(H, X)) complete. Theorem ∗ ∀ H ⊂ c0, k · k1-closed norming. Then (c0, w(c0, H)) locally complete, hence (c0, µ(c0, H)) is complete.

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 Another example: James’ space

Theorem J James space, H ⊂ J∗ k · k-closed norming subspace. Then (J, µ(J, H)) is complete.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Theorem X SWCG. Then (X ∗, µ(X ∗, X)) analytic ⇔ X separable.

The Mackey topology and coarser Mackey topologies The dual Mackey topology The Mackey topology on the dual of a Banach space: SWCG Banach spaces

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 on the dual of a Banach space: SWCG Banach spaces

Theorem X SWCG. Then (X ∗, µ(X ∗, X)) analytic ⇔ X separable.

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 on the dual of a Banach space: a question of Kakol, Kubis, and M. López-Pellicer

Question [Kakol–Kubis–López-Pellicer, 2011]: X separable. Is it true (X ∗, µ(X ∗, X)) analytic?

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space f K 1 U1 U2 1/2 U3 1/3 Z U4

K Z 0

The Mackey topology and coarser Mackey topologies The dual Mackey topology A result of R. B. Kirk, 1973

Theorem K compact. Then every zero subset of K is µ(M(K ), C(K ))-open.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space K

U1 U2 U3 Z U4

The Mackey topology and coarser Mackey topologies The dual Mackey topology A result of R. B. Kirk, 1973

Theorem K compact. Then every zero subset of K is µ(M(K ), C(K ))-open.

f 1

1/2 1/3

K Z 0

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 result of R. B. Kirk, 1973

Theorem K compact. Then every zero subset of K is µ(M(K ), C(K ))-open.

f K 1 U1 U2 1/2 U3 1/3 Z U4

K Z 0

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 result of R. B. Kirk, 1973

K

U1 U2 U3 Z U4

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 result of R. B. Kirk, 1973

K g1 1 U1 U2 U3 Z U4

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 result of R. B. Kirk, 1973

K g1 1 U1 U2 g2 0 1 U3 Z U4

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 result of R. B. Kirk, 1973

K g1 1 U1 U2 g2 0 1 U3 0 1 Z U4 g3

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 result of R. B. Kirk, 1973

K g1 1 U1 U2 g2 0 1 U3 0 1 Z U4 g3 1 0 g4

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 result of R. B. Kirk, 1973

Theorem (Kirk, 1973) K compact, 1st-countable. Then (K , µ(M(K ), C(K ))) discrete.

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 answer to the question of Kakol–Kubis–López-Pellicer

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Appendix References ReferencesI

M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler. Banach Space Theory: the Basis for Linear and Non-Linear Analysis Springer-Verlag, New York, 2011. M. Fabian, V. Montesinos, and V. Zizler, On weak compactness in L1 spaces. Rocky J. Math. 39, 6 (2009), 1885–1893. J. Bonet and B. Cascales, Non-complete Mackey topologies on Banach spaces. Bull. Australian Math. Soc., 2010. G. Schlüchtermann and R. Wheeler, The Mackey dual of a Banach space Note di Matematica, 11 (1991), 273–287.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space Appendix References ReferencesII

A.S. Granero and M. Sánchez, The class of universally Krein–Šmulyan spaces Bull. London Math. Soc. 39, 4 (2007), 529–540 R. B. Kirk A note on the Mackey topology for (Cb(X)∗, Cb(X)). Pacific J. Math., 45, 2, (1973). J. Kakol, W. Kubis, and M. López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis. Springer-Verlag 2011.

A. J. Guirao, V. Montesinos Some remarks on the Mackey topology in a Banach space