UDC 513.88+513.83

MATHEMATICS

V. S. RETAKH

ON SUBSPACES OF A COUNTABLE INDUCTIVE LIMIT

(Presented by Academician A. N. Kolmogorov, 20 III 1970)

Let \(E=\lim_{\to} E_n\) be an inductive limit of locally convex spaces and let \(H\) be a subspace of \(E\) (in the topology induced from \(E\)). We shall say that \(H\) has property \((T_0)\) if \(H'=(\lim_{\to}(E_n\cap H))'\), and property \((T)\) if \(H=\lim_{\to}(E_n\cap H)\). Obviously, \((T)\) implies \((T_0)\). We shall give necessary and sufficient conditions for the fulfillment of properties \((T_0)\) and \((T)\) under certain assumptions concerning \(E\).

1. Let \(G=\lim_{\to} G_n\) be an inductive limit of locally convex spaces. We shall say that \(G\) has property \((M_0)\) if all \(G_n\) are of countable type* and contain an absolutely convex neighborhood of zero \(U_n\) such that \(U_n\subset U_{n+1}\) and

\[ \forall i\ \exists j:\ \forall k>j\ \forall f\in G_j'\ \forall \varepsilon>0\ \exists g\in G_k':\ |f(x)-g(x)|<\varepsilon\ \forall x\in U_i. \tag{*} \]

If condition \((*)\) is replaced by the following: \(\forall i\ \exists j:\ \forall k>j\) the topologies induced on \(U_i\) from \(G_j\) and \(G_k\) coincide, then we shall say that \(G\) has property \((M)\). \((M)\) implies \((M_0)\). If \(E=\lim_{\to}E_n\) is a strict inductive limit of spaces of countable type, then \(E\) has property \((M)\).

Theorem 1. If

\[ E/H=\lim_{\to} E_n/E_n\cap H \]

has property \((M_0)\), then \(H\) has property \((T_0)\). If \(E\) has property \((M_0)\), or \(E\) is a strict inductive limit** and all \(E_n/E_n\cap H\) are of countable type, then the converse assertion is also true.

Theorem 2. If \(E/H\) has property \((M)\), then \(H\) has property \((T)\). If \(E\) has property \((M)\), or \(E\) is a strict inductive limit** and all \(E_n/E_n\cap H\) are of countable type, then the converse assertion is also true.

Remark. According to \((^7)\), if \(E\) is a strict inductive limit of Fréchet–Schwartz spaces, then for a closed \(H\) property \((T_0)\) is equivalent to property \((T)\).

For the proof of Theorems 1 and 2 one must use the scheme of the proofs of Theorem 2 from \((^3)\) and Theorem \(1'\) from \((^4)\), and Theorem 3 below.

We shall give the example, constructed by V. P. Palamodov and the author, of a closed subspace \(H\) of the space \(D((-1,2))\) not having property \((T_0)\). It differs from the example in \((^6)\) by its simplicity and by the fact that \(H\) is the image of an operator from the space \(D\).

Let \(\varphi\in D(R)\), with \(\operatorname{supp}\varphi=[1,2]\). Define a mapping

\[ Q:D((-1,0))\to D((-1,2)) \]

by the formula \(Q(f)=f+f*\varphi\). Put \(H=\operatorname{Im} Q\),

\[ E_n=D\left(\left[-1+2^{-n},\,2-2^{-n}\right]\right),\qquad H_n=E_n\cap H. \]

It is easy to see that \(Q\) is injective and \(H\) is closed. Suppose that \(H\) has property \((T_0)\). Since \(E\) is a strict—

* I.e., they possess a countable fundamental system of neighborhoods of zero.
** In this case it is not assumed that all \(E_n\) are of countable type.

inductive limit, then by Theorem 1 there exist a neighborhood of zero \(V_3\) in \(E_3\) and \(j>3\) such that

\[ \forall u\in E'_j,\quad u|_{H_j}=0\quad \forall \varepsilon>0\ \exists v\in E'_{j+1},\quad v|_{H_{j+1}}=0:\ |u(x)-v(x)|<\varepsilon\ \forall x\in V_3 . \]

Obviously \(\delta_{-2^{-j}}|_{H_j}=0\). Let \(v\in E'_{j+1}\) and \(v|_{H_{j+1}}=0\). The latter means that
\(v(f+f*\varphi)=0\), \(\forall f\in D([-1+2^{-j-1},-2^{-j-1}])\), whence it follows that \(v\) is infinitely differentiable on \([-1+2^{-j-1},2^{-j-1}]\). But \(\delta_{-2^{-j}}\) cannot be approximated by infinitely differentiable functions on \(V_3\). We have arrived at a contradiction; consequently, \(H\) does not have property \((T_0)\).

Let us note that it follows from this that \(Q^{-1}\) is discontinuous.

From Theorem 2, with the aid of the \(3\times 3\)-lemma for the category of locally convex spaces \((^2)\), there follows the main result of \((^5)\). However, instead of Theorem 2 it suffices to use the weaker Theorem 1 from \((^3)\).

II. Let \(L\) be the category of vector spaces over the field \(\mathbf C\) or \(\mathbf R\). By \(\widetilde L\) we shall denote the category of inverse spectra over \(L\) with the natural series \(\mathbf N\), ordered increasingly, as the common set of indices.

V. P. Palamodov in \((^1)\) studied the derived functors of the left exact functor
\(\operatorname{Pro}:\widetilde L\to L\), which assigns to each inverse spectrum
\(\mathscr X=\{X_p,\alpha_p^q:X_q\to X_p\}\) from \(\widetilde L\) its projective limit. He found a necessary and sufficient condition for \(\operatorname{Pro}^1\mathscr X\) to vanish in the case when the spaces \(X_p\) \((p\in\mathbf N)\) can be endowed with the topologies of Fréchet spaces so that the mappings \(\alpha_p^q\) are continuous.

We shall prove an analogous theorem for the dual, in a certain sense, case.

Definition. An absolutely convex set \(B\) in a vector space \(E\) will be called a Banach disk if the space \(E_B\), spanned by \(B\) and endowed, as a seminorm, with its Minkowski functional, is Banach.

Theorem 3. Let in each space \(X_p\) of the inverse spectrum
\[ \mathscr X=\{X_p,\alpha_p^q\} \]
from \(\widetilde L\) there exist a sequence of Banach disks
\[ (B_p^k)_{k\in\mathbf N} \]
such that
\[ \bigcup_{k=1}^{\infty} B_p^k=X_p \]
and
\[ [(\alpha_p^{p+1})^{-1}(B_p^k)]\cap B_{p+1}^r \]
is a Banach disk for arbitrary \(p,k,r\). Then \(\operatorname{Pro}^1\mathscr X=0\) if and only if in each \(X_p\) there exists a Banach disk \(B_p\) such that
\[ \alpha_p^{p+1}(B_{p+1})\subset B_p \]
and
\[ \forall i\ \exists j:\ \alpha_i^j(X_j)\subset B_i+\alpha_i(\operatorname{Pro}\mathscr X). \]

For the proof we shall need a lemma which in essence belongs to Banach.

Lemma. Let \(X,Y\) be vector spaces and \(f:X\to Y\) a linear mapping. Suppose that the sequences of absolutely convex sets \((U_n)_{n\in\mathbf N}\) and \((V_n)_{n\in\mathbf N}\) in \(X\) and \(Y\), respectively, form fundamental systems of neighborhoods of zero for the topologies of separable additive groups. If the group \(X\) is complete, the graph of \(f\) is closed, and \(f(U_n)\) is, for every \(n\), of second category in \(Y\), then \(f\) is open.

Proof of Theorem 3. Sufficiency follows from Theorem 11.2 in \((^1)\). We shall prove necessity.

Put
\[ \Pi=\prod_{p=1}^{\infty} X_p . \]
According to \((^1)\), if \(\operatorname{Pro}^1\mathscr X=0\), then the mapping
\[ \pi:\Pi\to\Pi \]
defined by the formula
\[ \pi(x_1,x_2,x_3,\ldots)=(x_1-\alpha_1^2(x_2),\,x_2-\alpha_2^3(x_3),\ldots) \tag{**} \]
is epimorphic. Introduce in \(\Pi\) the topology \(\varphi\) of an additive group with a fundamental system of neighborhoods of zero \((V_n)_{n\in\mathbf N}\), where
\[ V_n=0\times\cdots\times 0\times X_n\times X_{n+1}\times\cdots \]
(the \(0\) occurring \(n-1\) times). Since \((\Pi,\varphi)\) is complete and \(\pi\) is epimorphic, for each \(p\) най-

there exists \(B_n^k\) such that \(\pi(B_1^{k_1}\times \ldots \times B_m^{k_m}\times X_{m+1}\times X_{m+2}\times \ldots)\) is of the second category in \((\Pi,\varphi)\), for every \(m\). Introduce on \(\Pi\) the topology \(\tau\) of a group with a fundamental system of neighborhoods of zero \((U_n)_{n\in\mathbb N}\), where

\[ U_n=2^{-n}B_1^{k_1}\times \ldots \times 2^{-n}B_n^{k_n}\times X_{n+1}\times X_{n+2}\times \ldots . \]

By the lemma, \(\pi:(\Pi,\tau)\to(\Pi,\varphi)\) is open. Hence \(\forall i\ \exists j:\)

\[ \pi(B_1^{k_1}\times \ldots \times B_i^{k_i}\times X_{i+1}\times X_{i+2}\times \ldots)\subset (0\times \ldots 0\times X_j\times 0\times \ldots). \]

To complete the proof one must use formula (**).

I thank V. P. Palamodov and D. A. Raikov for their attention to this work.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
19 III 1970

REFERENCES

1. V. P. Palamodov, Matem. sborn., 75 (117), No. 4, 567 (1968).
2. D. A. Raikov, DAN, 188, No. 5, 1006 (1969).
3. V. S. Retakh, DAN, 184, No. 1, 44 (1969).
4. V. S. Retakh, Funkts. analiz, 3, No. 4, 63 (1969).
5. V. Ptak, Czech. Math. J., 19 (94), No. 3, 547 (1969).
6. W. Stowikowski, Bull. Acad. Pol. sci., 13, No. 6, 415 (1965).
7. M. de Wilde, Mém. Soc. R. des Sci. de Liège, 18, No. 2 (1969).