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Acta Mathematica Sinica, English Series
Apr., 2011, Vol. 27, No. 4, pp. 737–740
Published online: March 15, 2011
DOI: 10.1007/s10114-011-8540-1
Http://www.ActaMath.com
Acta Mathematica Sinica,
English Series
Springer-Verlag Berlin Heidelberg &
The Editorial Office of AMS 2011
Mackey First Countability and Docile Locally Convex Spaces
Carlos BOSCH GIRAL
Departamento de Matemáticas, ITAM, Rı́o Hondo #1,
Col. Progreso Tizapan, México, DF 01080, Mexico
Email : [email protected]
Thomas E. GILSDORF
Department of Mathematics, University of North Dakota,
Grand Forks, ND 58202–8376, USA
Email : [email protected]
Claudia GÓMEZ-WULSCHNER
Departamento de Matemáticas, ITAM, Rı́o Hondo #1,
Col. Progreso Tizapan, México, DF 01080, Mexico
Email : [email protected]
Abstract We define a generalization of Mackey first countability and prove that it is equivalent to
being docile. A consequence of the main result is to give a partial affirmative answer to an old question
of Mackey regarding arbitrary quotients of Mackey first countable spaces. Some applications of the
main result to spaces such as inductive limits are also given.
Keywords
Bounded set, docile space, Mackey first countable, inductive limit
MR(2000) Subject Classification
1
46A03, 46A13
Introduction
Throughout this paper space refers to a Hausdorff locally convex space E = (E, τ ) over K
(K = R or C). In 1945, Mackey [1] published one of his fundamental papers on topological
vector spaces, defining numerous concepts. One of those concepts is what we now call Mackey
first countable (see [1, p. 182]): Given any sequence (An ) of bounded sets in E, there exists a
sequence (αn ) of nonzero scalars such that ∞
n=1 αn An remains bounded. Ka̧kol and Saxon [2]
defined in 2002 the concept of a docile locally convex space: E is docile if every subspace
F of infinite dimension contains an infinite-dimensional bounded set; that is, F contains a
bounded set B in which there exists an infinite collection of linearly independent elements.
Every metrizable space is both Mackey first countable and docile. In [2] it is shown that the
property of being docile is also weaker than the Fréchet Urysohn property. See also [3]. Kummet
showed in [4] that every Mackey first countable space is docile. In this paper, we will define a
generalization of Mackey first countability and prove that this new property and that of being
Received November 17, 2008, accepted April 9, 2009
Bosch C., et al.
738
docile are equivalent. The last section of the paper is devoted to various properties of such
spaces, in particular, for inductive limits.
2
Main Result
Definition 1 A locally convex space E is sequentially Mackey first countable if given any
∞
sequence (An ) of bounded sets in E, and any sequence (xk ) from n=1 An , there exists a
sequence (αk ) of scalars with infinitely many nonzero terms such that {αk xk : k ∈ N} is bounded.
Clearly every Mackey first countable space is sequentially Mackey first countable. We do
not know if sequentially Mackey first countable implies Mackey first countable. The main result
is the next.
Theorem 1 A Hausdorff locally convex space is docile if and only if it is sequentially Mackey
first countable.
Proof The proof that any sequentially Mackey first countable space is docile follows the same
way as in [4], and for completeness we reproduce it here. Suppose E is sequentially Mackey first
countable and let F be any infinite-dimensional subspace of E. Then F contains a collection
{xn : n ∈ N} of linearly independent elements. Using {xn : n ∈ N} as a sequence of bounded
sets, there exists a sequence (αk ) of scalars with infinitely many nonzero terms such that
{αk xk : k ∈ N} is bounded. Thus, F contains an infinite-dimensional bounded set and we
conclude that E is docile.
Conversely, suppose that E is not sequentially Mackey first countable. Let (An ) denote
a sequence of bounded sets in E that does not satisfy the definition of sequential Mackey
∞
first countability and let (xk ) denote a sequence of elements from n=1 An such that for any
sequence (αk ) of scalars having infinitely many nonzero terms, {αk xk : k ∈ N} is unbounded.
We first observe that dim ({xk : k ∈ N}) = ∞. For if not, then {xk : k ∈ N} would be contained
in a finite dimensional subspace of E. Without loss of generality, assume that {xk : k ∈ N} is
linearly independent. Let F = span ({xk : k ∈ N}), the linear span of {xk : k ∈ N}. Clearly,
dim (F ) = ∞. Suppose B is any infinite-dimensional subset of F . Then B contains a collection
of the form, {βkm xkm : m ∈ N}, for some sequence (βkm : m ∈ N) of nonzero scalars. By our
assumption, such a collection cannot be bounded. Hence, E is not docile.
3
Some Applications of Main Result
We now collect some results on the properties of docile spaces. It has been observed in [2]
and [5, p. 73] that the continuous linear image of a docile space is docile. Thus, we obtain the
following partial solution to an old question of Mackey (see [1, p. 188]) regarding whether an
arbitrary quotient of a Mackey first countable space is Mackey first countable.
Corollary 2 An arbitrary quotient of a Mackey first countable space is sequentially Mackey
first countable.
Next, we consider inductive limits and spaces having a fundamental sequence of bounded
sets. From [6, 8.3, p. 248], a sequence (Bn ) of bounded sets in a space E is called a fundamental
sequence of bounded sets (f.s.b.), if given any bounded set A of E, there exists an m ∈ N such
that A ⊂ Bm . Without loss of generality, we assume that each Bn is absolutely convex, and
that Bn ⊂ Bn+1 .
Mackey First Countability and Docile L.C.S.
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Let E1 ⊂ E2 ⊂ · · · be a sequence of spaces such that En ⊂ En+1 and with continuous
injections, id : En → En+1 for each n ∈ N. Denote by E = indn (En ) the locally convex
inductive limit of this sequence. The inductive limit is proper if En En+1 for each n. Recall
[6, Definition 8.5.11, p. 285] that E is α-regular if every bounded subset of E is contained in some
step En , and E is regular if every bounded subset of E is contained in and bounded in some step
En . Some recent results on regular inductive limits can be found in [7]. In Theorems 4.2 (resp.
Theorem 4.3) of [5] Ka̧kol, Saxon and Todd show that the weak dual of a barrelled metrizable
space (resp. the strong dual of a metrizable space) is docile if and only if the space is normable.
In our next result we show that no proper α-regular inductive limit is docile, and we generalize
Theorems 4.2 and 4.3 of [5] to spaces having an f.s.b. in which no member is absorbing.
Theorem 3 (a) A proper α-regular inductive limit cannot be docile.
(b) Part (a) does not extend to general inductive limits.
(c) Suppose E has an f.s.b. (Bn ) such that no Bn is absorbing in E. Then E is not docile.
(d) A barrelled space with an f.s.b. is docile if and only if it is normed. In particular, no
proper (LB)-space is docile.
Proof (a) If E is a proper α-regular inductive limit, then we may choose xk ∈ Ek+1 \Ek , for
each k ∈ N. If E were sequentially Mackey first countable, then there would exist a sequence
(αk ) of scalars with infinitely many nonzero terms such that {αk xk : k ∈ N} is E-bounded.
The regularity of E would then imply that {αk xk : k ∈ N} is contained in some step En , an
impossibility.
(b) Concerning the existence of metrizable (LF )-spaces, see [6, Section 8.7]: Because every
metrizable space is docile, we know that part (a) does not extend to general inductive limits.
(c) Pick x1 in E such that x1 ∈ span {B1 }. Continue in this way, forming a sequence
(xk : k ∈ N) in E and a subsequence (Bnk : k ∈ N) of (Bn ), such that xk ∈ span {Bnk }, k =
1, 2, . . . . Now let (αk ) be any sequence of scalars having an infinite number of nonzero terms, and
suppose {αk xk : k ∈ N} is bounded in E. Then there exists m ∈ N such that {αk xk : k ∈ N} ⊂
Bm . By choosing k sufficiently large such that nk > m, αk = 0, and xk ∈ span {Bm }, we arrive
at a contradiction. E cannot be docile.
(d) Suppose the barrelled space E has an f.s.b. (Bn ) and is not normed. If even one of
the sets Bn were absorbing in E, then its closure would be a bounded neighborhood of zero,
an impossibility. Thus, the argument of part (c) applies, and E is not docile. In the case of a
proper (LB)-space E, recall that such a space is barrelled. For each n ∈ N, let
E
Bn = Kn ,
the E-closure of the unit ball Kn of En . The nonmetrizability of E (see [6, 8.5.18, p. 288])
implies that no Bn is absorbing in E. Now apply essentially the same argument as in (c), by
using the fact ([6, 8.5.20, p. 289, and 8.1.22, p. 232]) that if B is any bounded subset of E, then
E
for some p ∈ N, B ⊂ pUp where Up is a bounded zero neighborhood in Ep .
Note that any (gDF )-space ([6, 8.3.1 (ii), p. 248]) with an f.s.b. (Bn ) for which no Bn is
absorbing in E cannot be docile.
Two of the several possible generalizations of metrizable spaces are spaces having compatible
webs (see [6, 9.1.42]), and spaces satisfying the Mackey convergence condition (or strict Mackey
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Bosch C., et al.
condition); see [6, Definition 5.1.29, p. 158]. In our final observations below, we indicate that
the property of being Mackey first countable or docile is not in general, related to these types
of spaces.
Proposition 4 (a) Regular (LB)-spaces are webbed and not docile.
(b) For p > 1, (lp , σ (lp , lq )), (1/p) + (1/q) = 1, is a webbed, Mackey first countable space
that does not satisfy the Mackey convergence condition.
(c) A strict (in the sense of [8]) (LB)-space is webbed, satisfies the strict Mackey condition,
but is not docile.
Acknowledgements T. Gilsdorf would like to acknowledge research support for this paper
as part of a Fulbright Garcı́a Robles Scholarship at the Instituto Tecnológico Autónomo de
México (ITAM), Mexico City, 2006–2007. He is also grateful to the University of North Dakota
for Developmental Leave support. C. Bosch and C. Gómez are partially supported by the
Asociación Mexicana de Cultura, A. C.
References
[1] Mackey, G. W.: On infinite-dimensional linear spaces. Trans. Amer. Math. Soc., 66(57), 155–207 (1945)
[2] Kakol, J., Saxon, S. A.: Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex
topology. J. London Math. Soc., 66, 388–406 (2002)
[3] Kakol, J., Saxon, S. A., Todd, A. R.: Pseudocompact spaces X and df-spaces Cc (X). Proc. Amer. Math.
Soc., 132(6), 1703–1712 (2004)
[4] Kummet, C.: Bounded Sets in Locally Convex Spaces, Master’s Thesis, University of North Dakota, 2005
[5] Kakol, J., Saxon, S. A., Todd, A. R.: Docile Locally Convex Spaces, Contemp. Math., Vol. 341, Amer.
Math. Soc., Providence, RI, 2004, 73–77
[6] Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces, North Holland Math. Studies, Vol. 131,
1987
[7] Qiu, J. H.: Characterizations of weakly sequentially retractive (LM)-spaces. Acta Mathematica Sinica,
Chinese Series, 49(6), 1231–1238 (2006)
[8] Horváth, J.: Topological Vector Spaces and Distributions, Vol. 1, Academic Press, New York, 1966