BANACH-MACKEY SPACES

In recent publications the concepts of fast completeness and local barrel edness have been shown to be related to the property of all weak-* bounded subsets of the dual (of a locally convex space) being strongly bounded. In this paper we clarify those relationships, as well as giving several different characterizations of this property.

pointed out that the "only if" part is not correct and proposed a notion "locally barreled" which is weaker than fast completeness.They proved that if E is locally barreled, then all o(E" ,E)-bounded sets are /(E' ,E)-bounded.They also formulated a certain property (P) and shoed that E is locally barreled If it satisfies property (P) and if all o(E" ,E)-bounded sets are /9(E' ,E)-bounded.Thus, hen E has property (P), a necessary and sufficient condition for all G(E" ,E)-bounded sets to be /(E' ,E)-bounded is for E to be locally barreled.In [3] (3ilsdorf proved that henever E is locally barreled, then the families of eakly and strongly bounded subsets of II(E,F) are identical and that, when property (P) holds, the two statements are equi val ent.
In the present paper we give a number of necessary and sufficient conditions, as well as some sufficient conditions, for eak-.bounded subsett to be ttrongly bounded, and then investigate the relationships between them.In particular, we prove that whenever all G(E' ,E)-bounded sets are /9(E' ,E)-bounded and F is any locally convex space, then the families of eakly and strongly bounded subsets f II(E,F) are identlcal ,which yields the above-cited result of [3] as a direct consequence).
Each Frchet space Is locally barreled, but we present below an example of barreled space which i not locally barreled.This shows that all weak-* bounded sets ,av be strongly bounded without a space being locally barreled.
"t turns out that property (P is rather demanding.In particular Le how that f all weak-, bounded sets are strongly bounded, that prooerty P) .oesnot hold unless each linear unctional is continuous (which is never the case, for instance, n an nfnite dimensional Frchet space).Thus there are even many Banach spaces (which are lcally barreled and fast complete) which do not have property (P 2. BANACH-MACKEY SPACES (VAR OUS DESCR PT IONS).
In [I]-[3] some conditions for eak-.bounded sets to be strongly bounded have been nvestigated.
For brevity, we denote a Hausdorff locally convex linear topological space by the abbreviation l.c.s.As in [4], Def.10-4-3, e call a 1.c.s.
begn by giving a number of necessary and sufficient conditions for weak-, bounded sets o/ be strongly bounded.
The following statements are pairwise equi val ent: (EoT is a Banach-Mackey space; ($3) each barrel in (E,3"} is a bornivore (absorbs bounded sets in ($4) t?(E",E' IE Ig(E,E'); ($5) or any absolutely convex, bounded, closed subset B of E, the topology on the linear hull E B of B generated by the Minkoski functional PB of B is finer than that topology /9(E,E restricted to EB; (S) for any 1.c.s.F and any family bounded subsets o E covering E, a subset of the space (E,F) (of continuous linear operators from E to F) is pointwise bounded if and only if it is bounded on each element o S (S-bounded).
Next e sho that ($3) is equivalent to ($5).Let B be as in ($5) and let W be a barrel in (E3") (so that EBrI is a typical neighborhood of 0 for the topology relativized to EB).To kno that W will lways absorb B is to kno that some positive scalar-multiple of the set {x E: PB (X _<1} is contained in 14, which is to kno that ($5) holds.
That (Sa) implies ($2) is trivial.14e shall complete the proof by assuming that ($3) holds and demonstrating that (Sa) follos.Let then F and S be as in (S), let A be an element o S, let B be a pointNise bounded subset o (E,F), and let p be a continuous seminorm on F. The set B{xE:p(T(x))_<l (trdl)} is a barrel in E nd by ($3), B absorbs bounded subsets of E in particular B absorbs A. It follows that I is bounded on A, which establishes ($6).Q.E.D.
In the present section e give some sufficient conditions for weak-* bounded sets to be strongly bounded.if EB,PB) is a Banach ace, then B is called a Bapach disk and E is said be ast =omolete if each bounded subset of E is contained in a bounded Banach disk: b> i (,pB) is a barreled normed pace, then B is called a barreled dis and i said t be lcal barreled i each bunded subset o E is contained in a closed ouned barreled disk.
HEOREM 2. Let (E,) be a l.c.s.Each o he ollowing statements implies that E is a Banach-Mackey space: or each absolutely summable sequence c o scalars and each null sequence in (E,) the series x is convergent OOF. e irst sho that (1)  We conclude by shoing that (3) implies ($3).Suppose that (3) holds and assu that there eists a barrel W in (E,) ich is not a bonivore.
Then there is a bounded sequence .:yn}such that yn# each n and so {xnn/n} is a null euence in the complement o W. Deine T E by or each c.I /', then Mp{I/(y n) i:n is inite and or all c,d have #(T(c))-/(T(d))i_ < n=Z= (cn-dn)/s (Yn/n) < M-n=Z=i(csn-dn)/nl (3.2) It ollows that T is (,)-(E,E') ctinus bounded subsets o .Since the closed unit ball D is (,-compact, it ollos that te image T(D) is (E,E" )-compact and so (E,E' )-bnded as ell.But W (being a barrel) th absorbs T(D) hich contains the range o the sequence x as a subset: absurd Thus, holds.Q.E.D.
TOREM 3. I (E,) has the convex compactness poperty, then condition (3) Th eo em 2 s sat s ied.P. Assume that (E,) has the convex coactness property and let :( and c be as in (3) o eorem 2. Then the range A o x i$ compact and so the clos solute convex hull [A] is compact.Let d be the sequence c divided by its -norm.Then each o the partial su Zmd x is in [A] and so there exists a limit point o the partial sums.For to inite su md x and Z*d x and any COROLLARY.I a l.c.s.(E,) is either at complete or has the convex compactness pety, th it is a Banach-Mackey space.

COUNTER-EXAMPLES.
In this section we show that none ef the c=nditins of Theorem 2 is necessary E t be a Banacn-Mackey space.

EXAMPLE
(a barreled Banach-Mckev .pacewhich is not locally barreled).It -hown in [] that there ex.ists a HausOorf inductive limit .E,J' of CrOchet oaces .'.E such that each bounded subset of E is contained in some E but that .here_ists closed, absolutely convex, bounded subset B of E which is not bounded in any pace E Obviously (E,3") is barreled and so is a Banach-Mackey space as well.Assume that (E,J') were locally barreled.Then we may also assume that the set above is such that E B is a barreled space.Denote by #B the Minkowski functional and let mN be such that EBC Let {U be a nested neighborhood base of 0 in E consisting of closed, absolutely convex sets.Then [J :nN} generates a metri=able locally conve topology 3"o on E B which is finer than the norm topology induced by B. It follows that (EB'o) is complete ([9] I.l.).It now follows from the generali:ed .-losedgraph theorem ([4] Th. 12-5-7) that J is exactly the norm topology induced o 0 B. Wence each set n BIJ contains a positive multiple of B, whence follows that /ach U absorbs B. This means that B is bounded in E absurd.
e note that the fact that (E.) is not locally barreled can also be deduced from [4] T. I?-5-10.EXAMPLE 2 (a Banach-Mackey space which is not semi-reflexive).Choose any Banacn space which is not semi-reflexive.
EXAMPLE 3 (a Banach-Mackey space which does not have property (3) of Theorem 2).
It is shown in [5] (cf.also [] 31.6) that there exists a Hausdorff inductive limit (E,J') of Frchet spaces (E ,J containing a bounded sequence {y not contained in any of the spaces E Assume that (3) of Theorem 2 holds.As in the last paragraph of the proof of Theorem 2, we have that T(D) is (E,E )-co,act and so a Banach disk in (E,O).By the localization theorem for strictly webbed spaces ([7] 35.), T(D) is contained in E or some m: absurd .BANACH-MACKEY SPACES AND PROPERTY P.
In [2], Koea and Bilsdorf formulated a property (P) and proved that if E has property (P) and if each (E ,E)-bounded set is /9(E ,E) -bounded then E is locally barreled.Thus if a space has property (P), it is a Banach-Mackey space if and only if it is locally barreled.We prove below that property (P) is actually quite demanding.For reference we set down this property: (P)   for each absolutely convex, bounded, closed subset B of E, there exists a barrel W in E such that BrE B.
The following result seemed rather surprising.
Then each linear unctional on E is continuous.PRDOF.Let B be any absolutely convex, bounded, closed subset of E and denote by PB the Minkowski functional of B. By Theorem ($6) it follows that the topology induced by PB is finer than the relativized /(E,E )-topology.But property implies IB is coarser as well--hence B is the relativized topology /(E,E ).
Assume there is a discontinuous linear functional on (E,3").Then there exists a DEFINITION (c.[2] or [3]).Let BI be a disk, E B the linear hull o B, and PB the MinkoNski unctional o B: ):n} i bounded, it ollo that th sequence o partial sum is Cauchy.Hce Ed x exists (and equals s).