COUNTABILITY CONDITIONS IN LOCALLY CONVEX SPACES

B. S. BRUDOVSKII

(Presented by Academician P. S. Novikov on May 8, 1963)

Djedonne, in paper (²), proved that if in a barrelled or bornological space \(E\) there exists a fundamental sequence of bicompact sets, then \(E\) is a dense subspace of the strong dual \(E_1\) to a Fréchet—Montel space. In this article Djedonne posed two questions:

Does \(E\) always coincide with \(E_1\)?

Will the stated result be true for infrabarrelled spaces?

In § 1 an affirmative answer is given to Djedonne’s questions. In § 2 locally convex spaces with a fundamental sequence of bounded sets are studied; it turns out that for these spaces the result of Djedonne stated above is true.

§ 1. A fundamental sequence of sets of a certain class \(\mathfrak A\) is a sequence of sets of this class such that every set from \(\mathfrak A\) is contained in some set of this sequence. A fundamental sequence of bicompact sets in a topological vector space will, for brevity, be called a \(K\)-sequence.

Lemma 1. If in a topological vector space \(E\) there exists a \(K\)-sequence, then every bounded set in \(E\) is relatively bicompact (i.e. has bicompact closure).

Proof. Let \(A\) be a bounded but not relatively bicompact set, and let \((K)_ {n \in N}\) be a \(K\)-sequence in \(E\). Then, whatever \(n\) may be, \(A \not\subset nK_n\) (since \(x \to nx\) is a homeomorphism of \(E\) onto \(E\), and therefore \(nK_n\) is bicompact). Consequently, for each \(n\) there exists \(x_n \in A\) such that \(x_n \notin nK_n\). Since \(A\) is bounded, \(x_n/n \to 0\), and therefore the set \(K=\{x_n/n,0\}_{n \in N}\) is bicompact. By assumption, then \(K \subset K_p\) for some \(p\). But this is impossible, since \(x_p \notin pK_p\), whence \(x_p/p \notin K_p\).

A locally convex space is called infrabarrelled*, if in it every convex set absorbing all bounded sets is a neighborhood of zero. All barrelled and all bornological spaces are infrabarrelled.

Theorem 1. An infrabarrelled space \(E\) in which there exists a \(K\)-sequence is the strong dual to a Fréchet—Montel space.

Proof. By the lemma, the topology of compact convergence \(T_k\) on \(E'\) coincides with the strong topology \(\beta(E',E)=\beta\), and, moreover, \((E',\beta)=(E',T_k)\) is metrizable. Since \(E\) is infrabarrelled, every bounded set of \((E',\beta)\) is equicontinuous and therefore (since \(T_k=\beta\)) relatively bicompact. Consequently, \((E',\beta)\) is quasi-complete and hence, by virtue of its metrizability, complete. Thus \((E',\beta)\) is a complete metrizable space with relatively bicompact bounded sets, i.e. a Fréchet—Montel space. Thus

* (¹), Ex. 12 to § 2, Ch. III. Concerning the properties used of the strong dual to an infrabarrelled space, see (¹), Ex. 6 to § 3, Ch. IV.

as in \(E\) every bounded set is relatively bicompact (Lemma 1), \(E\) is semireflexive, i.e. \((E', \beta)' = E\). It remains to note that since \(E\) is infrabarreled, its topology coincides with \(\beta(E, E')\). Thus, \(E\) is the strong dual of the Fréchet—Montel space \((E', \beta)\).

Corollary. Every infrabarreled space in which there exists a \(K\)-sequence is complete.

Proof. The strong dual of a Fréchet space is complete.

§ 2. For brevity, a fundamental sequence of closed bounded sets in a topological vector space will be called a \(P\)-sequence.

Lemma \(1'\). If in a topological vector space \(E\) there exists a \(P\)-sequence, then in \(E\) every bounded set is closed bounded.

The proof is analogous to the proof of Lemma 1.

Lemma 2. If \(E\) is an infrabarreled space in which there exists a \(P\)-sequence, then every bounded set from its completion \(\hat E\) is contained in the completion of some closed bounded set from \(E\).

Proof. By Lemma \(1'\), every bounded set from \(E\) is closed bounded. As in the proof of Theorem 1, one can show that \((E', T_p)\), where \(T_p\) is the topology of uniform convergence on closed bounded sets, is a Fréchet—Montel space and, consequently, is both barrelled and boundedly closed simultaneously. From the latter it follows that \(E\) is regular and \(\hat E \subseteq E''\)*. Let \(A\) be a bounded set in \(\hat E\); then \(A\) is bounded in \((\hat E, \sigma(\hat E, E'))\), and therefore also in \((E'', \sigma(E'', E'))\). By the regularity of \(E\), there exists a bounded set \(K \subset E\), \(\sigma(E'', E')\)-dense in \(A\). Since \(K \subset \hat E\) and the topology \(\sigma(\hat E, E')\) is induced on \(\hat E\) by the topology \(\sigma(E'', E')\), it follows that \(\overline{K}^{(\hat E,\sigma(\hat E,E'))}\) (the \(\sigma(\hat E, E')\)-closure of \(K\) in \(\hat E\)) contains \(A\). But \(\overline{K}^{(\hat E,\sigma(\hat E,E'))} = \overline{K}^{\hat E}\), and \(\overline{K}^{E} = \overline{K}\); hence \(\hat K \supset A\). It remains to add that, by Lemma \(1'\), \(K\) is closed bounded in \(E\).

Remark. If in an infrabarreled space \(E\) there exists a \(P\)-sequence, then in \(\hat E\) there exists a \(K\)-sequence. Indeed, it follows from Lemma 2 that if \((P)_n{}_{n\in N}\) is a \(P\)-sequence in \(E\), then \((\hat P_n)\) is a \(K\)-sequence in \(\hat E\).

Theorem 2. If \(E\) is an infrabarreled space in which there exists a \(P\)-sequence, then \(\hat E\) is the strong dual of a Fréchet—Montel space.

Proof. Since \(E\) is infrabarreled, \(\hat E\) is barrelled. At the same time, by the remark to Lemma 2, in \(\hat E\) there exists a \(K\)-sequence, so it remains to apply Theorem 1.

In conclusion we shall show that in the space \(c_0\) of numerical null sequences, endowed with the topology \(T\) induced from \((m, \tau(m,l^1))\), where \(\tau(m,l^1)\) is the Mackey topology in the space \(m\) of bounded sequences, there is a \(P\)-sequence, but there is no \(K\)-sequence. Indeed, in \(l^1\) every weakly bicompact set is strongly bicompact, so that on every equicontinuous set from \(m\) the topologies \(\tau(m,l^1)\) and \(\sigma(m,l^1)\) coincide. Consequently, the balls

\[ B_n=\{(x_i)\in m:\sup_{i\in N}|x_i|\le n\} \]

in \(m\), being equicontinuous sets, are bicompact in \((m,\tau(m,l^1))\). Since they form a fundamental sequence of bounded sets in \((m,\beta(m,l^1))\), and every bicompact set in \((m,\tau(m,l^1))\) is bounded in

* See (1), exercise 10 to § 3 of Chapter IV. An infrabarreled space is called regular if every bounded set from \((E'', \sigma(E'',E'))\) is contained in the closure (in the topology \(\sigma(E'',E')\)) of some bounded set from \(E\).

\((m, \beta(m,l^1))\), the balls \(B_n\) form a \(K\)-sequence in \((m,\tau(m,l^1))\). But \(B_n\) is the closure in \((m,\tau(m,l^1))\) of the ball

\[ A_n=\{(x_i)\in c_0:\sup_{i\in N}|x_i|\le n\} \]

from \(c_0\), and, since \((m,\tau(m,l^1))\) is complete, \((\hat c_0,\hat T)=(m,\tau(m,l^1))\). Thus the balls \(A_n\) in \((c_0,T)\) are quite bounded and form a \(P\)-sequence. If there were now a \(K\)-sequence in \((c_0,T)\), then (by Lemma 1) the balls \(A_n\) in \((c_0,T)\) would be relatively bicompact, and then their closures in \((\hat c_0,\hat T)\) would be complete; but the completion of the ball \(A_n\) in the topology \(T\) is the ball \(B_n\), and it does not belong to \(c_0\).

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
29 IV 1963

CITED LITERATURE

¹ N. Bourbaki, Topological Vector Spaces, IL, 1958. ² T. Dieudonné, Proc. Am. Math. Soc., 8, 367 (1957).