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(LF)-spaces with more-than-separable quotients
J. Math. Anal. Appl. 434 (2016) 12–19
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
(LF )-spaces with more-than-separable quotients Stephen A. Saxon 1 Department of Mathematics, University of Florida, PO Box 118105, Gainesville, FL 32611-8105, USA
a r t i c l e
i n f o
Article history: Received 11 August 2015 Available online 5 September 2015 Submitted by J. Bonet In memory of Professor John Horváth Keywords: Properly separable quotients (LF)-spaces Primitive spaces
a b s t r a c t The separable quotient problem remains open for Banach spaces. The recent solution for barrelled spaces counters Saxon–Narayanaswami’s solution for (LF)-spaces. W.J. Robertson proposed properly separable quotients. We fully answer her quartercentury-old question, proving (LF)-spaces admit properly separable quotients with rare exception. Corollary: Every (LF)-space except ϕ admits properly separable quotients if and only if Banach’s classic separable quotient problem has a positive solution. © 2015 Elsevier Inc. All rights reserved.
1. Introduction We follow Horváth [7], deviations noted. Locally convex topological vector spaces (lcs’s) and their quotients are assumed to be Hausdorff over the scalar field of reals or complexes. A lcs (E, τ ) is, respectively, an (LF )-, (LB)-, or (LN )-space if it is the inductive limit of a strictly increasing sequence {(En , τn )}n of Fréchet, Banach, or normed spaces where the relative topology τn+1 |En induced on En is coarser than τn for each n ∈ N. Such is a defining sequence for (E, τ ) with steps (En , τn ) (n ∈ N). Every subsequence is also a defining sequence. Other places call (E, τ ) a proper (LF )-, (LB)-, or (LN )-space, as En is required to be a proper subspace of En+1 . If each τn+1 |En = τn , then each τ |En = τn and (E, τ ) is a strict (LF )-, (LB)-, or (LN )-space. If F is a subspace of some En , we may conveniently write (F, τn ) or (F, τ ) to indicate F with the relative τn |F or τ |F topology, respectively. The classic separable quotient problem asks if all infinite-dimensional Banach spaces admit infinitedimensional separable quotients. As we just learned, some barrelled spaces do not [8]. However, Saxon and Narayanaswami [15] proved (A) Every (LF )-space admits an infinite-dimensional separable quotient.
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E-mail address: [email protected]. Thanks to Profs. Jerzy Kąkol and Aaron Todd for helpful personal communications.
http://dx.doi.org/10.1016/j.jmaa.2015.09.005 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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Robertson [9] defines a topological vector space (tvs) to be properly separable if it has a proper dense ℵ0 -dimensional subspace. A Hausdorff tvs is properly separable if and only if it has a proper dense subspace which is separable. Trivially, properly separable implies infinite-dimensional and separable; the converse holds under metrizability. However, the strong dual ϕ of the Fréchet space ω of all scalar sequences is ℵ0 -dimensional, hence separable, but is not properly separable since it bears its strongest locally convex topology and admits no proper dense subspaces. Likewise for the quotients: ϕ is a strict (LB)-space having infinite-dimensional separable quotients, but no properly separable quotients. Rosenthal [10] proved that for any infinite compact set X, either 2 or c is an [infinite-dimensional separable] quotient of the Banach space Cc (X). Eidelheit [4, Satz 2] showed that ω is a [properly separable] quotient of every non-normable Fréchet space. Our prequel [8] shows that ω is a quotient of every non-Banach barrelled space of the form Cc (X). We saw that a GM -space never has properly separable quotients, but has infinite-dimensional separable quotients precisely when it is an Sσ space (i.e., is the union of an increasing sequence of proper closed subspaces). Robertson [9] showed that If some Fréchet step of a strict (LF )-space E is non-normable, then E admits a properly separable quotient isomorphic to ω. We relax the hypothesis to If an (LF )-space E is not an (LN )-space, and give the converse, all with shorter proof a la Eidelheit. Our main results (see Abstract) benefit from a recently-completed study [18] of weak barrelledness. 2. Some elements of weak barrelledness An absorbing balanced convex closed set in a lcs E is a barrel. Every lcs E has a base of neighborhoods of 0 consisting of barrels, and E is barrelled if each barrel is a neighborhood of 0. Barrelled spaces embody the Uniform Boundedness Principle fundamental to functional analysis, for, via the bipolar theorem, a lcs E is barrelled if and only if every pointwise bounded (σ (E , E)-bounded) subset of the continuous dual E is equicontinuous; i.e., is uniformly bounded on some neighborhood of 0 in E. If V is a barrel in E, then {nV }n covers E, and nV = 12 nV − 12 nV is nowhere dense if V is not a neighborhood of 0. The Baire category theorem thus ensures that every Fréchet space is barrelled. A lcs E is semi-Baire-like (sBL) if, whenever it is covered by an increasing sequence {An }n of balanced convex closed sets, some An is absorbing, a barrel. If E is barrelled and sBL, it is Baire-like (BL). If E is barrelled and non-Sσ , it is quasi-Baire (QB). These terms were formally introduced in [11]; earlier, Amemiya and K¯omura [1] had proved that every metrizable barrelled space is BL. Trivially, BL ⇒ QB ⇒ barrelled, BL ⇒ sBL, and QB ⇒ non-Sσ . A lcs E is inductive (in [6], weakly barrelled) if it is the inductive limit of every increasing sequence {En }n of subspaces that cover E; is ϕ-complemental if every algebraic complement of each closed ℵ0 -codimensional subspace is a topological complement with its strongest locally convex topology (and hence isomorphic to ϕ); is primitive (in [19], has property f |Ln ) if a linear form is continuous whenever it has continuous restrictions to each member of some increasing covering sequence of subspaces. It is easily seen in [18] that non-Sσ ⇒ inductive; barrelled ⇒ inductive ⇒ ϕ-complemental ⇒ primitive; and, under metrizability, non-Sσ ⇔ primitive. The following table, from the first and fifth rows of Figs. 2 and 3 in [18], is a convenient summary. Boxes 3 and 5 indicate equivalences under metrizability; boxes 1, 2, and 4 in [18] are omitted here. 3) metrizable
BL ⇓ sBL
⇒
QB
⇒
⇓ non-Sσ
⇒
barrelled
⇒
⇓ inductive
⇒
ϕ-complemental ⇒ primitive
5) metrizable
A sequence of scalars is eventually zero if all but finitely many of its members are zero. For each sequence {fn }n in the dual E of a lcs E, we define the associated eventually zero subspace of E, denoted ez {fn }n ,
S.A. Saxon / J. Math. Anal. Appl. 434 (2016) 12–19
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by writing ez {fn }n = {x ∈ E : {fn (x)}n is eventually zero.}. The special case ez {fn }n = E is well-used in [5,18]. We state two elementary results from [18, Section 2]. (P1) A lcs E is an Sσ space if and only if there is a closed ℵ0 -codimensional subspace. (P2) A lcs E is primitive if and only if linear forms which vanish on a closed countable-codimensional subspace of E must be continuous. Saxon–Wilansky [20] proved (B) A Banach space admits an infinite-dimensional separable quotient if and only if it contains a dense non-barrelled subspace. We observed [8, Theorem 2.2] the similar result (C) A lcs admits an infinite-dimensional separable quotient if and only if it contains a dense Sσ subspace. Let us prove Theorem 1. The following three assertions about a lcs E are equivalent. (i) E admits a properly separable quotient E/M . (ii) E contains a dense non-primitive subspace F . (iii) There exists {fn }n ⊂ E with ez {fn }n dense and proper in E. Proof. (i) ⇒ (iii). By hypothesis, there exists a proper dense subspace G containing M with dim G/M = ℵ0 . Let {xn }n be a cobasis for M in G and choose y ∈ E\G. For each n ∈ N the Hahn–Banach theorem ensures fn ∈ E such that fn (M ) = {0}, fn (xn ) = 1 = fn (y), and fn (xi ) = 0 if 1 ≤ i < n. Since ez {fn }n contains M and {xn }n , it contains G, therefore is dense, and is also proper, since it does not contain y. (iii) ⇒ (ii). For Mn = i≥n fi⊥ , it is always true that G = ez {fn }n is the union of the increasing sequence {Mn }n of closed subspaces and each dim Mn+1 /Mn ≤ 1. Therefore dim G/M1 ≤ ℵ0 . By hypothesis, G is not closed; thus dim G/M1 = ℵ0 . Let y ∈ E\G, put F = G + sp {y}, and define h ∈ F ∗ by writing h (y) = 1 and h (G) = {0}. Then h ∈ / F , yet h vanishes on M1 , which proves F is not primitive via (P2). (ii) ⇒ (i). By (P2) there exist a closed ℵ0 -codimensional subspace N in F and f ∈ F ∗ \F with f (N ) = {0}. Let M be the closure of N in E. Now f ⊥ + M /M is an ℵ0 − 1 = ℵ0 -dimensional space that is proper and dense in (F + M ) /M , thus in E/M ; the latter is properly separable. 2 As dense subspaces of GM -spaces are barrelled [3], hence primitive, they never admit properly separable quotients. But the familiar Fréchet spaces admit dense non-barrelled primitive subspaces (dominated by Fréchet spaces), as do the spaces Cc (X) with X infinite. To the latter two classes applies Theorem 2. (See [8, Theorem 2.3].) If a non-Sσ space E admits an infinite-dimensional separable quotient, then it admits a properly separable quotient. Proof. By (C) and (P1), some dense subspace F has a closed ℵ0 -codimensional subspace M . Now F + M = E, since E is non-Sσ . Therefore F + M /M is a proper dense ℵ0 -dimensional subspace of E/M . 2 The result fails if non-Sσ is relaxed to a weaker property in the above table, as evidenced by the barrelled, hence inductive space ϕ. However, countable-codimensional subspaces and direct sums preserve said properties [18], yielding Theorem 3. A lcs F admits properly separable quotients if F × ϕ does. Proof. By Theorem 1, the space F × ϕ contains a dense non-primitive subspace G. Now H = G + ({0} × ϕ) is not primitive, since its countable-codimensional subspace G is not. Since H is the direct sum of
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H (F × {0}) and {0} × ϕ, the former space is non-primitive, and clearly dense in F × {0}; Theorem 1 applies. 2 Corollary 1. (See [9, Example].) If some infinite-dimensional Banach space F lacks a properly separable quotient, then so does some strict (LB)-space E ≈ ϕ, namely, E = F × ϕ. From (A) and Theorem 2 we have Theorem 4. All quasi-Baire (LF )-spaces admit properly separable quotients. This leaves the (LF )-spaces that are not QB, characterized as the (LF )-spaces having no dense steps, defined as (LF )1 -spaces [15,16]. The (LF )2 -spaces are characterized as those (LF )-spaces that are QB and not BL, and (LF )3 -spaces as those that are BL. Since the (LF )3 -spaces coincide with the metrizable (LF )-spaces [16,17], they obviously admit properly separable quotients via (A). Theorem 4 simply says all (LF )2 or 3 -spaces do. Perhaps Robertson [9] chose to study strict (LF )-spaces as exemplars of (LF )1 -spaces. Even some (LF )1 -spaces are implicitly covered by Theorem 4. Theorem 5. If some step En of an (LF )-space E satisfies dim E/E n = ℵ0 with E n ⊂ Ej (j = 1, 2, . . . ), then E admits a properly separable quotient. Proof. Being barrelled, E is ϕ-complemental, thus isomorphic to E n ×ϕ, so E n is a quotient of E. According to [16, Theorem 10], E n is an (LF )-space with dense step En , hence an (LF )2 or 3 -space with properly separable quotient. Since the taking of quotients is transitive, the result follows. 2 3. When is ω a quotient of a given (LF)-space? A perfectly general answer supersedes [9, Theorem]. Theorem 6. Let {(En , τn )}n be a defining sequence of Fréchet spaces for an (LF )-space (E, τ ). The following three assertions are equivalent. (1) Some quotient of E is a copy of ω. (2) E is not an (LN )-space. (3) For some k ∈ N, no τk -neighborhood of 0 is τ -bounded. Proof. (1) ⇒ (2). Suppose (E, τ ) is the inductive limit of a sequence of normed spaces {(Fn , n )}n whose sequence of unit balls is increasing. As ω is a quotient of E, there is a sequence {fn }n ⊂ E such that, for every sequence {εn }n of positive scalars, {εn fn }n is not σ (E , E)-bounded. But for εn = 1/ (1 + fn |Fn n ), if x is in some Fk and n ≥ k, then |εn fn (x)| ≤ εn fn |Fn n x n ≤ x n ≤ x k , so that {εn fn }n is bounded at each point x ∈ E, a contradiction. (2) ⇒ (3). Suppose (3) is false. For each k ∈ N there exists a τ -bounded absolutely convex τk -neighborhood Uk of 0, and since the sum of two bounded sets is bounded, we may assume Uk ⊂ Uk+1 . Since τ is Hausdorff, the gauge of Uk is a norm k on Ek , and clearly (E, τ ) is the inductive limit of the sequence {(Ek , k )}k , which contradicts (2). (3) ⇒ (1). Let {Vn }n be a base of balanced neighborhoods of 0 for the Fréchet space (Ek , τk ) such that each Vn+1 + Vn+1 ⊂ Vn . By (3) we may inductively choose fn ∈ E such that f1 is unbounded on V1 ⊥ and each fn+1 is unbounded on Vn+1 {f1 , . . . , fn } . (The intersection, indeed, is not τ -bounded since its sum with a suitable n-dimensional τ -bounded set is a τk -neighborhood of 0.) Let (an )n be an arbitrary scalar sequence, a point in ω. Since f1 is unbounded on the balanced set V1 , we may choose x1 ∈ V1 such
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that f1 (x1 ) = a1 . We may likewise choose x2 ∈ V2 f1⊥ such that f2 (x2 ) = a2 − f2 (x 1 ). We inductively ⊥ continue, finding xn+1 ∈ Vn+1 {f1 , . . . , fn } such that fn+1 (xn+1 ) = an+1 − fn+1 for each i≤n xi n ∈ N. The series i xi converges to some x in the Fréchet space (Ek , τk ) and the restriction to Ek of each fn is τk -continuous, so f1 (x) = a1 and fn+1 (x) = fn+1 (x1 + · · · + xn + xn+1 ) = an+1 for n ≥ 1; i.e., (fn (x))n = (an )n . Therefore the linear mapping T : (E, τ ) → ω that takes x ∈ E into (fn (x))n ∈ ω is not only continuous, since the fn ’s are, but its restriction to the Fréchet space (Ek , τk ) is both continuous and onto ω. Thus the restriction is open, by Pták’s open mapping theorem, and then so is T itself. Therefore (E, τ ) /T −1 (0) is a copy of ω. 2 The last argument gives Eidelheit’s result [4, Satz 2] that ω is a quotient of every non-normable Fréchet space F . More exotic methods extend the result to countable-codimensional subspaces of F [2,12,13]. (Here, a set S is countable if |S| ≤ ℵ0 .) Some quotient of each (LF )3 -space is an infinite-dimensional separable Fréchet space [16, Theorem 9]. However, Corollary 2. Some quotient of an (LF )3 -space E is a copy of ω if (and only if) E is non-normable. Proof. Every Baire-like (LN )-space is normable. 2 4. Closing arguments We independently prove the main result, that all known (LF )-spaces except ϕ admit properly separable quotients. But we must first add Lemma 1. Let {fn }n and A be in the dual F of a Fréchet space F . If {fn }n is equicontinuous and G = sp A◦ is dense and proper in F , then ez {fn + hn }n is dense in F for some {hn }n ⊂ A. Proof. Let {Un }n be a base of closed balanced convex neighborhoods of 0 in F such that {fn }n ⊂ U1◦ and each Un+1 + Un+1 ⊂ Un . Since closed finite-codimensional subspaces L are always complemented, by hypothesis A is unbounded on each set of the form L G Un . Set M1 = 1 and arbitrarily choose x1 ∈ G U1 and h1 ∈ A. The definition of G implies sup{|h(x1 )| : ⊥ h ∈ A} = M2 is finite. Choose x2 ∈ {f1 , h1 } G U2 and h2 ∈ A such that h2 (x2 ) ≥ M2 + 4. Next we set M3 = sup
2
|h (xk )| : h ∈ A
k=1
and continue inductively to find {xn }n ⊂ G, {hn }n ⊂ A, and {Mn }n such that for all n ∈ N, ⊥ G Un+1 ; hn+1 (xn+1 ) ≥ Mn+1 + 4; and xn+1 ∈ {f1 , . . . , fn , h1 , . . . , hn } n
Mn+1 = sup |h (xk )| : h ∈ A . k=1
Set {gn }n = {fn + hn }n . Let y ∈ U1
A◦ and n ∈ N be given. We define
a1 = −gn+1 (y) /gn+1 (xn+1 )
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and, for j ≥ 2, aj = −gn+j (y + a1 xn+1 + · · · + aj−1 xn+j−1 ) /gn+j (xn+j ) . Now |a1 | ≤ 2/ (hn+1 (xn+1 ) − |fn+1 (xn+1 )|) ≤ 2/ (Mn+1 + 4 − 1) ≤ 1. Suppose j ≥ 2 and we know |ak | ≤ 1 for k = 1, . . . , j − 1. Then
u=
j−1
ak xn+k ∈ Un ⊂ U1 ,
k=1
and |aj | =
N D,
where N = |gn+j (y + u)| ≤ |fn+j (y)| + |fn+j (u)| + |hn+j (y)| + |hn+j (u)| ≤ 3 + Mn+j = (Mn+j + 4) − 1 ≤ hn+j (xn+j ) − |fn+j (xn+j )| ≤ |gn+j (xn+j )| = D.
Therefore aj , well-defined, satisfies |aj | ≤ 1. By the principle of mathematical induction, |ak | ≤ 1 for all k ∈ N, so that z=y+
∞
ak xn+k ∈ y + Un ,
k=1
where the series converges (absolutely) in the Fréchet space F . If j ∈ N and k > j, then gn+j (xn+k ) = 0 by construction. Thus for each j ∈ N, gn+j (z) = gn+j
y+
j
ak xn+k
,
k=1
so that gn+j (z) = 0 by definition of aj . Thus z ∈ ez {gi }i indeed, the closure ez {gi }i ⊃ sp (U1 A◦ ) = G. 2
(y + Un ), whence y ∈ ez {gi }i ⊃ U1
A◦ ;
Now we need not depend on (A) for a Proof of Theorem 4. One of the steps (Ep , τp ) is dense (and proper) in the QB (LF )-space (E, τ ). By the open mapping theorem, (Ep , τ ) is not barrelled, so there exists a barrel B in (Ep , τ ) whose closure B in E is not a neighborhood of zero in sp B, τ . By [14], sp B is uncountable-codimensional in E. For some j > p, then, there is a linearly independent null sequence {xn }n in the jth step (Ej , τj ) with sp B sp {xn }n = {0}. ◦ For Bn = B + i≤n ai xi : each |ai | ≤ 1 , the bipolar theorem provides un ∈ Bn with un (xn+1 ) > 1. Let F be the closure of sp B Ej + sp {xn }n in (Ej , τj ), making (F, τj ) a Fréchet space on which {un |F }n is non-equicontinuous. Thus for some y ∈ F there is a subset {ηn }n of {un }n with |ηn (y)| > n, so that if A = {ηn |F }n , then G = sp A◦ is proper (y ∈ / G) and dense in (F, τj ), since A is bounded at each point of B Ej {xn }n . Setting fn = 0, we apply Lemma 1 to obtain a subset {hn }n of {ηn }n with ez {hn }n ⊃ ez {hn |F }n = F ⊃ Ep , which is dense in E. Since y ∈ / ez {hn }n , Theorem 1 ensures E has a properly separable quotient. 2 Theorem 7. The (LF )-spaces without properly separable quotients are precisely those of the form F × ϕ, where F is a Banach space without properly separable quotients.
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Proof. Let (E, τ ) be an (LF )-space with Fréchet steps (En , τn ). For each subset S, let S denote the closure of S in (E, τ ). I. Suppose E ≈ F × ϕ, where F is a Fréchet space. If F is non-normable, then ω is a properly separable quotient of E by Eidelheit’s result. If F is a Banach space, Theorem 3 ensures E has a properly separable quotient if and only if F does. II. Suppose some dim E/E n = ℵ0 . Then E ≈ E n × ϕ with E n barrelled, and either (a) some En+j ⊃ E n , so that E n is a Fréchet space and case I. applies, or else (b) Theorem 5 provides a properly separable quotient of E. III. If some dim E/E n < ℵ0 , then some E n+j = E, which implies E is quasi-Baire and admits properly separable quotients by Theorem 4. IV. Finally, suppose each dim E/E n > ℵ0 . Passing to a subsequence and re-naming if necessary, we may assume that dim E2n / E2n E 2n−1 > ℵ0 for every n ∈ N. Clearly, each Dn = E2n E 2n−1 is closed in the Fréchet space (E2n , τ2n ), and in the latter we easily construct a closed subspace Fn ⊃ Dn such that (Fn , τ2n ) /Dn is properly separable. Thus {(Fn , τ2n )}n is a defining sequence for (E, τ ) and each Fn ⊂ Dn+1 . If ω is a quotient of E, then our desired conclusion holds. We assume ω is not, so that by Theorem 6, each (Fn , τ2n ) has a neighborhood of 0 that is τ -bounded. As (F1 , τ ) /D1 , also, is properly separable, there exists 0 = fn ∈ (F1 , τ ) such that ez {fn }n is dense and proper in (F1 , τ ). The Hahn–Banach theorem provides {fn }n ⊂ (F2 , τ ) with each fn |F1 = fn . Since (F2 , τ4 ) /D2 is properly separable and D2 is closed in (F2 , τ ), one easily chooses 0 = hn ∈ (F2 , τ ) with hn (D2 ) = {0} and ez {hn }n dense and proper in (F2 , τ4 ). Let V be a τ4 |F2 -neighborhood of 0 which is τ -bounded. Each bn = sup {|fn (v)| : v ∈ V } is finite and positive. Therefore b−1 n fn n is τ4 |F2 -equicontinuous, while {cn hn }n is not for some sequence {cn }n of positive scalars large enough that cn hn (xn ) > 1 for some null sequence {xn }n in (F2 , τ4 ). Set A = {cn hn }n , so that F2 = sp A◦ ⊃ ez {cn hn }n = ez {hn }n is dense in (F2 , τ4 ). The lemma yields {gn }n ⊂ (F2 , τ4 ) with each gi = b−1 i fi + ηi ∈ (F2 , τ ) for some ηi in the set {cn hn }n , and such that ez {gn }n is dense in (F2, τ4 ). Since bi ηi vanishes on D2 ⊃ F1 , each fi = bi gi extends fi to an element in (F2 , τ ) , and ez {fn }n = ez {gn }n is dense in (F2 , τ4 ). Also, ez {fn }n is proper in F2 , since F1 ez {fn }n = ez {fn }n is proper in F1 . We similarly find {fn }n ⊂ (F3 , τ ) such that each fn |F2 = fn and ez {fn }n is dense and proper in F3 . Continuing recursively, we obtain fn n ⊂ E ∗ with fn |Fk n ⊂ (Fk , τ ) for each k ∈ N, so that fn n ⊂ (E, τ ) by primitivity, and with Fk ez fn n dense and proper in (Fk , τ ). Therefore ez fn n is dense and proper in (E, τ ), ensuring a properly separable quotient. 2 (A) is an immediate corollary. We also have Proof of (B). If a Banach space E has a properly separable quotient, Theorem 1 ensures a dense nonprimitive, hence non-barrelled subspace. Conversely, if H is a dense non-barrelled subspace of E, then there is a σ (E , H)-bounded sequence {ηn }n and a point y ∈ E with each ηn (y) > n. Set A = {ηn }n and fn = 0 and apply the lemma (y ∈ / sp A◦ ⊃ H is dense) to obtain a subset {hn }n of A with ez {hn }n dense. Since y ∈ / ez {hn }n , Theorem 1 ensures a properly separable quotient. 2 To (B) and Theorem 1 we add Corollary 3. An (LF )-space E has a properly separable quotient if and only if it has a dense non-barrelled subspace. Proof. Any dense non-primitive space is non-barrelled, so Theorems 1 and 7 reduce the proof to the case where E = F × ϕ with F a Banach space lacking properly separable quotients. By (B), then, every dense
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subspace of F is barrelled. Thus so is every dense subspace of E, as shown by the argument of Theorem 3, with barrelled replacing primitive. 2 Robertson [9] asked: If E is the strict inductive limit of a sequence of Banach spaces, each with a (properly) separable quotient, has E a properly separable quotient? The answer is Yes. More generally, Corollary 4. If each step (Ej , τj ) of an arbitrary (LF )-space (E, τ ) has a properly separable quotient, then so has E. Proof. Suppose E does not. Then Theorem 7 says (E, τ ) is a strict (LB)-space with Banach steps (Fk , τ ), where (F1 , τ ) has no properly separable quotients and each dim Fk+1 /Fk = 1. By Grothendieck’s equivalence theorem (see [17]) there exist j, k ∈ N such that (Ej , τj ) is between (F1 , τ ) and (Fk , τ ), with the inclusions continuous. Since dim Fk /F1 = k − 1 is finite, the interpolation (Ej , τj ) = (Ej , τ ) is closed in (E, τ ) ≈ (Ej , τj ) × ϕ. Since (Ej , τj ), a quotient of E, has a properly separable quotient, so does E, contradicting the supposition. 2 Corollary 5. Every (LF )-space except ϕ admits properly separable quotients if and only if every infinitedimensional Banach space does. Robertson’s secondary (LM )-space question prompts us to ask if there are non-complete metrizable barrelled spaces without properly separable quotients? References [1] I. Amemiya, Y. K¯ omura, Über nicht-vollständige Montel Räume, Math. Ann. 177 (1968) 273–277. [2] P. Pérez Carreras, J. Bonet, Una nota sobre un resultado de Eidelheit, Collect. Math. 33 (1982) 195–199. [3] V. Eberhardt, W. Roelcke, Über einem Graphensatz für lineare abbildungen mit metrisierbarem Zielraum, Manuscripta Math. 13 (1974) 53–68. [4] M. Eidelheit, Zur Theorie der Systeme linearer Gleichungen, Studia Math. 6 (1936) 130–148. [5] J.C. Ferrando, Jerzy Kąkol, Stephen A. Saxon, Characterizing P-spaces X in terms of Cp (X), J. Convex Anal. 22 (4) (2015), in press. [6] J.C. Ferrando, L.M. Sánchez Ruiz, On sequential barrelledness, Arch. Math. 57 (1991) 597–605. [7] J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, 1966. [8] J. Kąkol, S.A. Saxon, A.R. Todd, Barrelled spaces with(out) separable quotients, Bull. Aust. Math. Soc. 90 (2014) 295–303. [9] W.J. Robertson, On properly separable quotients of strict (LF) spaces, J. Aust. Math. Soc. 47 (1989) 307–312. [10] H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from p (μ) to r (ν), J. Funct. Anal. 4 (1969) 176–214. [11] S.A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972) 87–106. [12] S.A. Saxon, Every countable-codimensional subspace of an infinite-dimensional [non-normable] Fréchet space has an infinite-dimensional Fréchet quotient [isomorphic to ω], Bull. Pol. Acad. Sci. 39 (1991) 161–166. [13] S.A. Saxon, All separable Banach spaces are quotients of any countable-codimensional subspace of 1 , Bull. Pol. Acad. Sci. 39 (1991) 167–173. [14] S.A. Saxon, Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971) 91–96. [15] S.A. Saxon, P.P. Narayanaswami, Metrizable (LF)-spaces, (db)-spaces and the separable quotient problem, Bull. Aust. Math. Soc. 23 (1981) 65–80. [16] S.A. Saxon, P.P. Narayanaswami, (LF)-spaces, quasi-Baire spaces, and the strongest locally convex topology, Math. Ann. 274 (1986) 627–641. [17] S.A. Saxon, P.P. Narayanaswami, Metrizable [normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math. 93 (1989) 1–16. [18] S.A. Saxon, L.M. Sánchez Ruiz, Reinventing weak barrelledness, preprint. [19] S.A. Saxon, I. Tweddle, The fit and flat components of barrelled spaces, Bull. Aust. Math. Soc. 51 (1995) 521–528. [20] S.A. Saxon, A. Wilansky, The equivalence of some Banach space problems, Colloq. Math. XXXVII (1977) 217–226.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
(LF )-spaces with more-than-separable quotients Stephen A. Saxon 1 Department of Mathematics, University of Florida, PO Box 118105, Gainesville, FL 32611-8105, USA
a r t i c l e
i n f o
Article history: Received 11 August 2015 Available online 5 September 2015 Submitted by J. Bonet In memory of Professor John Horváth Keywords: Properly separable quotients (LF)-spaces Primitive spaces
a b s t r a c t The separable quotient problem remains open for Banach spaces. The recent solution for barrelled spaces counters Saxon–Narayanaswami’s solution for (LF)-spaces. W.J. Robertson proposed properly separable quotients. We fully answer her quartercentury-old question, proving (LF)-spaces admit properly separable quotients with rare exception. Corollary: Every (LF)-space except ϕ admits properly separable quotients if and only if Banach’s classic separable quotient problem has a positive solution. © 2015 Elsevier Inc. All rights reserved.
1. Introduction We follow Horváth [7], deviations noted. Locally convex topological vector spaces (lcs’s) and their quotients are assumed to be Hausdorff over the scalar field of reals or complexes. A lcs (E, τ ) is, respectively, an (LF )-, (LB)-, or (LN )-space if it is the inductive limit of a strictly increasing sequence {(En , τn )}n of Fréchet, Banach, or normed spaces where the relative topology τn+1 |En induced on En is coarser than τn for each n ∈ N. Such is a defining sequence for (E, τ ) with steps (En , τn ) (n ∈ N). Every subsequence is also a defining sequence. Other places call (E, τ ) a proper (LF )-, (LB)-, or (LN )-space, as En is required to be a proper subspace of En+1 . If each τn+1 |En = τn , then each τ |En = τn and (E, τ ) is a strict (LF )-, (LB)-, or (LN )-space. If F is a subspace of some En , we may conveniently write (F, τn ) or (F, τ ) to indicate F with the relative τn |F or τ |F topology, respectively. The classic separable quotient problem asks if all infinite-dimensional Banach spaces admit infinitedimensional separable quotients. As we just learned, some barrelled spaces do not [8]. However, Saxon and Narayanaswami [15] proved (A) Every (LF )-space admits an infinite-dimensional separable quotient.
1
E-mail address: [email protected]. Thanks to Profs. Jerzy Kąkol and Aaron Todd for helpful personal communications.
http://dx.doi.org/10.1016/j.jmaa.2015.09.005 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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Robertson [9] defines a topological vector space (tvs) to be properly separable if it has a proper dense ℵ0 -dimensional subspace. A Hausdorff tvs is properly separable if and only if it has a proper dense subspace which is separable. Trivially, properly separable implies infinite-dimensional and separable; the converse holds under metrizability. However, the strong dual ϕ of the Fréchet space ω of all scalar sequences is ℵ0 -dimensional, hence separable, but is not properly separable since it bears its strongest locally convex topology and admits no proper dense subspaces. Likewise for the quotients: ϕ is a strict (LB)-space having infinite-dimensional separable quotients, but no properly separable quotients. Rosenthal [10] proved that for any infinite compact set X, either 2 or c is an [infinite-dimensional separable] quotient of the Banach space Cc (X). Eidelheit [4, Satz 2] showed that ω is a [properly separable] quotient of every non-normable Fréchet space. Our prequel [8] shows that ω is a quotient of every non-Banach barrelled space of the form Cc (X). We saw that a GM -space never has properly separable quotients, but has infinite-dimensional separable quotients precisely when it is an Sσ space (i.e., is the union of an increasing sequence of proper closed subspaces). Robertson [9] showed that If some Fréchet step of a strict (LF )-space E is non-normable, then E admits a properly separable quotient isomorphic to ω. We relax the hypothesis to If an (LF )-space E is not an (LN )-space, and give the converse, all with shorter proof a la Eidelheit. Our main results (see Abstract) benefit from a recently-completed study [18] of weak barrelledness. 2. Some elements of weak barrelledness An absorbing balanced convex closed set in a lcs E is a barrel. Every lcs E has a base of neighborhoods of 0 consisting of barrels, and E is barrelled if each barrel is a neighborhood of 0. Barrelled spaces embody the Uniform Boundedness Principle fundamental to functional analysis, for, via the bipolar theorem, a lcs E is barrelled if and only if every pointwise bounded (σ (E , E)-bounded) subset of the continuous dual E is equicontinuous; i.e., is uniformly bounded on some neighborhood of 0 in E. If V is a barrel in E, then {nV }n covers E, and nV = 12 nV − 12 nV is nowhere dense if V is not a neighborhood of 0. The Baire category theorem thus ensures that every Fréchet space is barrelled. A lcs E is semi-Baire-like (sBL) if, whenever it is covered by an increasing sequence {An }n of balanced convex closed sets, some An is absorbing, a barrel. If E is barrelled and sBL, it is Baire-like (BL). If E is barrelled and non-Sσ , it is quasi-Baire (QB). These terms were formally introduced in [11]; earlier, Amemiya and K¯omura [1] had proved that every metrizable barrelled space is BL. Trivially, BL ⇒ QB ⇒ barrelled, BL ⇒ sBL, and QB ⇒ non-Sσ . A lcs E is inductive (in [6], weakly barrelled) if it is the inductive limit of every increasing sequence {En }n of subspaces that cover E; is ϕ-complemental if every algebraic complement of each closed ℵ0 -codimensional subspace is a topological complement with its strongest locally convex topology (and hence isomorphic to ϕ); is primitive (in [19], has property f |Ln ) if a linear form is continuous whenever it has continuous restrictions to each member of some increasing covering sequence of subspaces. It is easily seen in [18] that non-Sσ ⇒ inductive; barrelled ⇒ inductive ⇒ ϕ-complemental ⇒ primitive; and, under metrizability, non-Sσ ⇔ primitive. The following table, from the first and fifth rows of Figs. 2 and 3 in [18], is a convenient summary. Boxes 3 and 5 indicate equivalences under metrizability; boxes 1, 2, and 4 in [18] are omitted here. 3) metrizable
BL ⇓ sBL
⇒
QB
⇒
⇓ non-Sσ
⇒
barrelled
⇒
⇓ inductive
⇒
ϕ-complemental ⇒ primitive
5) metrizable
A sequence of scalars is eventually zero if all but finitely many of its members are zero. For each sequence {fn }n in the dual E of a lcs E, we define the associated eventually zero subspace of E, denoted ez {fn }n ,
S.A. Saxon / J. Math. Anal. Appl. 434 (2016) 12–19
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by writing ez {fn }n = {x ∈ E : {fn (x)}n is eventually zero.}. The special case ez {fn }n = E is well-used in [5,18]. We state two elementary results from [18, Section 2]. (P1) A lcs E is an Sσ space if and only if there is a closed ℵ0 -codimensional subspace. (P2) A lcs E is primitive if and only if linear forms which vanish on a closed countable-codimensional subspace of E must be continuous. Saxon–Wilansky [20] proved (B) A Banach space admits an infinite-dimensional separable quotient if and only if it contains a dense non-barrelled subspace. We observed [8, Theorem 2.2] the similar result (C) A lcs admits an infinite-dimensional separable quotient if and only if it contains a dense Sσ subspace. Let us prove Theorem 1. The following three assertions about a lcs E are equivalent. (i) E admits a properly separable quotient E/M . (ii) E contains a dense non-primitive subspace F . (iii) There exists {fn }n ⊂ E with ez {fn }n dense and proper in E. Proof. (i) ⇒ (iii). By hypothesis, there exists a proper dense subspace G containing M with dim G/M = ℵ0 . Let {xn }n be a cobasis for M in G and choose y ∈ E\G. For each n ∈ N the Hahn–Banach theorem ensures fn ∈ E such that fn (M ) = {0}, fn (xn ) = 1 = fn (y), and fn (xi ) = 0 if 1 ≤ i < n. Since ez {fn }n contains M and {xn }n , it contains G, therefore is dense, and is also proper, since it does not contain y. (iii) ⇒ (ii). For Mn = i≥n fi⊥ , it is always true that G = ez {fn }n is the union of the increasing sequence {Mn }n of closed subspaces and each dim Mn+1 /Mn ≤ 1. Therefore dim G/M1 ≤ ℵ0 . By hypothesis, G is not closed; thus dim G/M1 = ℵ0 . Let y ∈ E\G, put F = G + sp {y}, and define h ∈ F ∗ by writing h (y) = 1 and h (G) = {0}. Then h ∈ / F , yet h vanishes on M1 , which proves F is not primitive via (P2). (ii) ⇒ (i). By (P2) there exist a closed ℵ0 -codimensional subspace N in F and f ∈ F ∗ \F with f (N ) = {0}. Let M be the closure of N in E. Now f ⊥ + M /M is an ℵ0 − 1 = ℵ0 -dimensional space that is proper and dense in (F + M ) /M , thus in E/M ; the latter is properly separable. 2 As dense subspaces of GM -spaces are barrelled [3], hence primitive, they never admit properly separable quotients. But the familiar Fréchet spaces admit dense non-barrelled primitive subspaces (dominated by Fréchet spaces), as do the spaces Cc (X) with X infinite. To the latter two classes applies Theorem 2. (See [8, Theorem 2.3].) If a non-Sσ space E admits an infinite-dimensional separable quotient, then it admits a properly separable quotient. Proof. By (C) and (P1), some dense subspace F has a closed ℵ0 -codimensional subspace M . Now F + M = E, since E is non-Sσ . Therefore F + M /M is a proper dense ℵ0 -dimensional subspace of E/M . 2 The result fails if non-Sσ is relaxed to a weaker property in the above table, as evidenced by the barrelled, hence inductive space ϕ. However, countable-codimensional subspaces and direct sums preserve said properties [18], yielding Theorem 3. A lcs F admits properly separable quotients if F × ϕ does. Proof. By Theorem 1, the space F × ϕ contains a dense non-primitive subspace G. Now H = G + ({0} × ϕ) is not primitive, since its countable-codimensional subspace G is not. Since H is the direct sum of
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H (F × {0}) and {0} × ϕ, the former space is non-primitive, and clearly dense in F × {0}; Theorem 1 applies. 2 Corollary 1. (See [9, Example].) If some infinite-dimensional Banach space F lacks a properly separable quotient, then so does some strict (LB)-space E ≈ ϕ, namely, E = F × ϕ. From (A) and Theorem 2 we have Theorem 4. All quasi-Baire (LF )-spaces admit properly separable quotients. This leaves the (LF )-spaces that are not QB, characterized as the (LF )-spaces having no dense steps, defined as (LF )1 -spaces [15,16]. The (LF )2 -spaces are characterized as those (LF )-spaces that are QB and not BL, and (LF )3 -spaces as those that are BL. Since the (LF )3 -spaces coincide with the metrizable (LF )-spaces [16,17], they obviously admit properly separable quotients via (A). Theorem 4 simply says all (LF )2 or 3 -spaces do. Perhaps Robertson [9] chose to study strict (LF )-spaces as exemplars of (LF )1 -spaces. Even some (LF )1 -spaces are implicitly covered by Theorem 4. Theorem 5. If some step En of an (LF )-space E satisfies dim E/E n = ℵ0 with E n ⊂ Ej (j = 1, 2, . . . ), then E admits a properly separable quotient. Proof. Being barrelled, E is ϕ-complemental, thus isomorphic to E n ×ϕ, so E n is a quotient of E. According to [16, Theorem 10], E n is an (LF )-space with dense step En , hence an (LF )2 or 3 -space with properly separable quotient. Since the taking of quotients is transitive, the result follows. 2 3. When is ω a quotient of a given (LF)-space? A perfectly general answer supersedes [9, Theorem]. Theorem 6. Let {(En , τn )}n be a defining sequence of Fréchet spaces for an (LF )-space (E, τ ). The following three assertions are equivalent. (1) Some quotient of E is a copy of ω. (2) E is not an (LN )-space. (3) For some k ∈ N, no τk -neighborhood of 0 is τ -bounded. Proof. (1) ⇒ (2). Suppose (E, τ ) is the inductive limit of a sequence of normed spaces {(Fn , n )}n whose sequence of unit balls is increasing. As ω is a quotient of E, there is a sequence {fn }n ⊂ E such that, for every sequence {εn }n of positive scalars, {εn fn }n is not σ (E , E)-bounded. But for εn = 1/ (1 + fn |Fn n ), if x is in some Fk and n ≥ k, then |εn fn (x)| ≤ εn fn |Fn n x n ≤ x n ≤ x k , so that {εn fn }n is bounded at each point x ∈ E, a contradiction. (2) ⇒ (3). Suppose (3) is false. For each k ∈ N there exists a τ -bounded absolutely convex τk -neighborhood Uk of 0, and since the sum of two bounded sets is bounded, we may assume Uk ⊂ Uk+1 . Since τ is Hausdorff, the gauge of Uk is a norm k on Ek , and clearly (E, τ ) is the inductive limit of the sequence {(Ek , k )}k , which contradicts (2). (3) ⇒ (1). Let {Vn }n be a base of balanced neighborhoods of 0 for the Fréchet space (Ek , τk ) such that each Vn+1 + Vn+1 ⊂ Vn . By (3) we may inductively choose fn ∈ E such that f1 is unbounded on V1 ⊥ and each fn+1 is unbounded on Vn+1 {f1 , . . . , fn } . (The intersection, indeed, is not τ -bounded since its sum with a suitable n-dimensional τ -bounded set is a τk -neighborhood of 0.) Let (an )n be an arbitrary scalar sequence, a point in ω. Since f1 is unbounded on the balanced set V1 , we may choose x1 ∈ V1 such
S.A. Saxon / J. Math. Anal. Appl. 434 (2016) 12–19
16
that f1 (x1 ) = a1 . We may likewise choose x2 ∈ V2 f1⊥ such that f2 (x2 ) = a2 − f2 (x 1 ). We inductively ⊥ continue, finding xn+1 ∈ Vn+1 {f1 , . . . , fn } such that fn+1 (xn+1 ) = an+1 − fn+1 for each i≤n xi n ∈ N. The series i xi converges to some x in the Fréchet space (Ek , τk ) and the restriction to Ek of each fn is τk -continuous, so f1 (x) = a1 and fn+1 (x) = fn+1 (x1 + · · · + xn + xn+1 ) = an+1 for n ≥ 1; i.e., (fn (x))n = (an )n . Therefore the linear mapping T : (E, τ ) → ω that takes x ∈ E into (fn (x))n ∈ ω is not only continuous, since the fn ’s are, but its restriction to the Fréchet space (Ek , τk ) is both continuous and onto ω. Thus the restriction is open, by Pták’s open mapping theorem, and then so is T itself. Therefore (E, τ ) /T −1 (0) is a copy of ω. 2 The last argument gives Eidelheit’s result [4, Satz 2] that ω is a quotient of every non-normable Fréchet space F . More exotic methods extend the result to countable-codimensional subspaces of F [2,12,13]. (Here, a set S is countable if |S| ≤ ℵ0 .) Some quotient of each (LF )3 -space is an infinite-dimensional separable Fréchet space [16, Theorem 9]. However, Corollary 2. Some quotient of an (LF )3 -space E is a copy of ω if (and only if) E is non-normable. Proof. Every Baire-like (LN )-space is normable. 2 4. Closing arguments We independently prove the main result, that all known (LF )-spaces except ϕ admit properly separable quotients. But we must first add Lemma 1. Let {fn }n and A be in the dual F of a Fréchet space F . If {fn }n is equicontinuous and G = sp A◦ is dense and proper in F , then ez {fn + hn }n is dense in F for some {hn }n ⊂ A. Proof. Let {Un }n be a base of closed balanced convex neighborhoods of 0 in F such that {fn }n ⊂ U1◦ and each Un+1 + Un+1 ⊂ Un . Since closed finite-codimensional subspaces L are always complemented, by hypothesis A is unbounded on each set of the form L G Un . Set M1 = 1 and arbitrarily choose x1 ∈ G U1 and h1 ∈ A. The definition of G implies sup{|h(x1 )| : ⊥ h ∈ A} = M2 is finite. Choose x2 ∈ {f1 , h1 } G U2 and h2 ∈ A such that h2 (x2 ) ≥ M2 + 4. Next we set M3 = sup
2
|h (xk )| : h ∈ A
k=1
and continue inductively to find {xn }n ⊂ G, {hn }n ⊂ A, and {Mn }n such that for all n ∈ N, ⊥ G Un+1 ; hn+1 (xn+1 ) ≥ Mn+1 + 4; and xn+1 ∈ {f1 , . . . , fn , h1 , . . . , hn } n
Mn+1 = sup |h (xk )| : h ∈ A . k=1
Set {gn }n = {fn + hn }n . Let y ∈ U1
A◦ and n ∈ N be given. We define
a1 = −gn+1 (y) /gn+1 (xn+1 )
S.A. Saxon / J. Math. Anal. Appl. 434 (2016) 12–19
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and, for j ≥ 2, aj = −gn+j (y + a1 xn+1 + · · · + aj−1 xn+j−1 ) /gn+j (xn+j ) . Now |a1 | ≤ 2/ (hn+1 (xn+1 ) − |fn+1 (xn+1 )|) ≤ 2/ (Mn+1 + 4 − 1) ≤ 1. Suppose j ≥ 2 and we know |ak | ≤ 1 for k = 1, . . . , j − 1. Then
u=
j−1
ak xn+k ∈ Un ⊂ U1 ,
k=1
and |aj | =
N D,
where N = |gn+j (y + u)| ≤ |fn+j (y)| + |fn+j (u)| + |hn+j (y)| + |hn+j (u)| ≤ 3 + Mn+j = (Mn+j + 4) − 1 ≤ hn+j (xn+j ) − |fn+j (xn+j )| ≤ |gn+j (xn+j )| = D.
Therefore aj , well-defined, satisfies |aj | ≤ 1. By the principle of mathematical induction, |ak | ≤ 1 for all k ∈ N, so that z=y+
∞
ak xn+k ∈ y + Un ,
k=1
where the series converges (absolutely) in the Fréchet space F . If j ∈ N and k > j, then gn+j (xn+k ) = 0 by construction. Thus for each j ∈ N, gn+j (z) = gn+j
y+
j
ak xn+k
,
k=1
so that gn+j (z) = 0 by definition of aj . Thus z ∈ ez {gi }i indeed, the closure ez {gi }i ⊃ sp (U1 A◦ ) = G. 2
(y + Un ), whence y ∈ ez {gi }i ⊃ U1
A◦ ;
Now we need not depend on (A) for a Proof of Theorem 4. One of the steps (Ep , τp ) is dense (and proper) in the QB (LF )-space (E, τ ). By the open mapping theorem, (Ep , τ ) is not barrelled, so there exists a barrel B in (Ep , τ ) whose closure B in E is not a neighborhood of zero in sp B, τ . By [14], sp B is uncountable-codimensional in E. For some j > p, then, there is a linearly independent null sequence {xn }n in the jth step (Ej , τj ) with sp B sp {xn }n = {0}. ◦ For Bn = B + i≤n ai xi : each |ai | ≤ 1 , the bipolar theorem provides un ∈ Bn with un (xn+1 ) > 1. Let F be the closure of sp B Ej + sp {xn }n in (Ej , τj ), making (F, τj ) a Fréchet space on which {un |F }n is non-equicontinuous. Thus for some y ∈ F there is a subset {ηn }n of {un }n with |ηn (y)| > n, so that if A = {ηn |F }n , then G = sp A◦ is proper (y ∈ / G) and dense in (F, τj ), since A is bounded at each point of B Ej {xn }n . Setting fn = 0, we apply Lemma 1 to obtain a subset {hn }n of {ηn }n with ez {hn }n ⊃ ez {hn |F }n = F ⊃ Ep , which is dense in E. Since y ∈ / ez {hn }n , Theorem 1 ensures E has a properly separable quotient. 2 Theorem 7. The (LF )-spaces without properly separable quotients are precisely those of the form F × ϕ, where F is a Banach space without properly separable quotients.
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Proof. Let (E, τ ) be an (LF )-space with Fréchet steps (En , τn ). For each subset S, let S denote the closure of S in (E, τ ). I. Suppose E ≈ F × ϕ, where F is a Fréchet space. If F is non-normable, then ω is a properly separable quotient of E by Eidelheit’s result. If F is a Banach space, Theorem 3 ensures E has a properly separable quotient if and only if F does. II. Suppose some dim E/E n = ℵ0 . Then E ≈ E n × ϕ with E n barrelled, and either (a) some En+j ⊃ E n , so that E n is a Fréchet space and case I. applies, or else (b) Theorem 5 provides a properly separable quotient of E. III. If some dim E/E n < ℵ0 , then some E n+j = E, which implies E is quasi-Baire and admits properly separable quotients by Theorem 4. IV. Finally, suppose each dim E/E n > ℵ0 . Passing to a subsequence and re-naming if necessary, we may assume that dim E2n / E2n E 2n−1 > ℵ0 for every n ∈ N. Clearly, each Dn = E2n E 2n−1 is closed in the Fréchet space (E2n , τ2n ), and in the latter we easily construct a closed subspace Fn ⊃ Dn such that (Fn , τ2n ) /Dn is properly separable. Thus {(Fn , τ2n )}n is a defining sequence for (E, τ ) and each Fn ⊂ Dn+1 . If ω is a quotient of E, then our desired conclusion holds. We assume ω is not, so that by Theorem 6, each (Fn , τ2n ) has a neighborhood of 0 that is τ -bounded. As (F1 , τ ) /D1 , also, is properly separable, there exists 0 = fn ∈ (F1 , τ ) such that ez {fn }n is dense and proper in (F1 , τ ). The Hahn–Banach theorem provides {fn }n ⊂ (F2 , τ ) with each fn |F1 = fn . Since (F2 , τ4 ) /D2 is properly separable and D2 is closed in (F2 , τ ), one easily chooses 0 = hn ∈ (F2 , τ ) with hn (D2 ) = {0} and ez {hn }n dense and proper in (F2 , τ4 ). Let V be a τ4 |F2 -neighborhood of 0 which is τ -bounded. Each bn = sup {|fn (v)| : v ∈ V } is finite and positive. Therefore b−1 n fn n is τ4 |F2 -equicontinuous, while {cn hn }n is not for some sequence {cn }n of positive scalars large enough that cn hn (xn ) > 1 for some null sequence {xn }n in (F2 , τ4 ). Set A = {cn hn }n , so that F2 = sp A◦ ⊃ ez {cn hn }n = ez {hn }n is dense in (F2 , τ4 ). The lemma yields {gn }n ⊂ (F2 , τ4 ) with each gi = b−1 i fi + ηi ∈ (F2 , τ ) for some ηi in the set {cn hn }n , and such that ez {gn }n is dense in (F2, τ4 ). Since bi ηi vanishes on D2 ⊃ F1 , each fi = bi gi extends fi to an element in (F2 , τ ) , and ez {fn }n = ez {gn }n is dense in (F2 , τ4 ). Also, ez {fn }n is proper in F2 , since F1 ez {fn }n = ez {fn }n is proper in F1 . We similarly find {fn }n ⊂ (F3 , τ ) such that each fn |F2 = fn and ez {fn }n is dense and proper in F3 . Continuing recursively, we obtain fn n ⊂ E ∗ with fn |Fk n ⊂ (Fk , τ ) for each k ∈ N, so that fn n ⊂ (E, τ ) by primitivity, and with Fk ez fn n dense and proper in (Fk , τ ). Therefore ez fn n is dense and proper in (E, τ ), ensuring a properly separable quotient. 2 (A) is an immediate corollary. We also have Proof of (B). If a Banach space E has a properly separable quotient, Theorem 1 ensures a dense nonprimitive, hence non-barrelled subspace. Conversely, if H is a dense non-barrelled subspace of E, then there is a σ (E , H)-bounded sequence {ηn }n and a point y ∈ E with each ηn (y) > n. Set A = {ηn }n and fn = 0 and apply the lemma (y ∈ / sp A◦ ⊃ H is dense) to obtain a subset {hn }n of A with ez {hn }n dense. Since y ∈ / ez {hn }n , Theorem 1 ensures a properly separable quotient. 2 To (B) and Theorem 1 we add Corollary 3. An (LF )-space E has a properly separable quotient if and only if it has a dense non-barrelled subspace. Proof. Any dense non-primitive space is non-barrelled, so Theorems 1 and 7 reduce the proof to the case where E = F × ϕ with F a Banach space lacking properly separable quotients. By (B), then, every dense
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subspace of F is barrelled. Thus so is every dense subspace of E, as shown by the argument of Theorem 3, with barrelled replacing primitive. 2 Robertson [9] asked: If E is the strict inductive limit of a sequence of Banach spaces, each with a (properly) separable quotient, has E a properly separable quotient? The answer is Yes. More generally, Corollary 4. If each step (Ej , τj ) of an arbitrary (LF )-space (E, τ ) has a properly separable quotient, then so has E. Proof. Suppose E does not. Then Theorem 7 says (E, τ ) is a strict (LB)-space with Banach steps (Fk , τ ), where (F1 , τ ) has no properly separable quotients and each dim Fk+1 /Fk = 1. By Grothendieck’s equivalence theorem (see [17]) there exist j, k ∈ N such that (Ej , τj ) is between (F1 , τ ) and (Fk , τ ), with the inclusions continuous. Since dim Fk /F1 = k − 1 is finite, the interpolation (Ej , τj ) = (Ej , τ ) is closed in (E, τ ) ≈ (Ej , τj ) × ϕ. Since (Ej , τj ), a quotient of E, has a properly separable quotient, so does E, contradicting the supposition. 2 Corollary 5. Every (LF )-space except ϕ admits properly separable quotients if and only if every infinitedimensional Banach space does. Robertson’s secondary (LM )-space question prompts us to ask if there are non-complete metrizable barrelled spaces without properly separable quotients? References [1] I. Amemiya, Y. K¯ omura, Über nicht-vollständige Montel Räume, Math. Ann. 177 (1968) 273–277. [2] P. Pérez Carreras, J. Bonet, Una nota sobre un resultado de Eidelheit, Collect. Math. 33 (1982) 195–199. [3] V. Eberhardt, W. Roelcke, Über einem Graphensatz für lineare abbildungen mit metrisierbarem Zielraum, Manuscripta Math. 13 (1974) 53–68. [4] M. Eidelheit, Zur Theorie der Systeme linearer Gleichungen, Studia Math. 6 (1936) 130–148. [5] J.C. Ferrando, Jerzy Kąkol, Stephen A. Saxon, Characterizing P-spaces X in terms of Cp (X), J. Convex Anal. 22 (4) (2015), in press. [6] J.C. Ferrando, L.M. Sánchez Ruiz, On sequential barrelledness, Arch. Math. 57 (1991) 597–605. [7] J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, 1966. [8] J. Kąkol, S.A. Saxon, A.R. Todd, Barrelled spaces with(out) separable quotients, Bull. Aust. Math. Soc. 90 (2014) 295–303. [9] W.J. Robertson, On properly separable quotients of strict (LF) spaces, J. Aust. Math. Soc. 47 (1989) 307–312. [10] H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from p (μ) to r (ν), J. Funct. Anal. 4 (1969) 176–214. [11] S.A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972) 87–106. [12] S.A. Saxon, Every countable-codimensional subspace of an infinite-dimensional [non-normable] Fréchet space has an infinite-dimensional Fréchet quotient [isomorphic to ω], Bull. Pol. Acad. Sci. 39 (1991) 161–166. [13] S.A. Saxon, All separable Banach spaces are quotients of any countable-codimensional subspace of 1 , Bull. Pol. Acad. Sci. 39 (1991) 167–173. [14] S.A. Saxon, Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971) 91–96. [15] S.A. Saxon, P.P. Narayanaswami, Metrizable (LF)-spaces, (db)-spaces and the separable quotient problem, Bull. Aust. Math. Soc. 23 (1981) 65–80. [16] S.A. Saxon, P.P. Narayanaswami, (LF)-spaces, quasi-Baire spaces, and the strongest locally convex topology, Math. Ann. 274 (1986) 627–641. [17] S.A. Saxon, P.P. Narayanaswami, Metrizable [normable] (LF)-spaces and two classical problems in Fréchet [Banach] spaces, Studia Math. 93 (1989) 1–16. [18] S.A. Saxon, L.M. Sánchez Ruiz, Reinventing weak barrelledness, preprint. [19] S.A. Saxon, I. Tweddle, The fit and flat components of barrelled spaces, Bull. Aust. Math. Soc. 51 (1995) 521–528. [20] S.A. Saxon, A. Wilansky, The equivalence of some Banach space problems, Colloq. Math. XXXVII (1977) 217–226.