Reports of the Academy of Sciences of the USSR
1969. Volume 184, No. 1

UDC 513.88:513.83

MATHEMATICS

V. S. RETAKH

ON THE DUAL OF A SUBSPACE OF A COUNTABLE INDUCTIVE LIMIT

(Presented by Academician A. N. Kolmogorov on 26 IV 1968)

Let (E=\lim\limits_{\to} E_n), where (E_n) are locally convex spaces, and let (H) be a vector subspace of (E), endowed with the topology induced from (E). There is a natural embedding
[
i:\ \lim_{\to}(E_n\cap H)\to H.
]
The question is posed: when is
[
i':\ H'\to\bigl(\lim_{\to}(E_n\cap H)\bigr)'
]
bijective?

The corollary of Proposition 1 and Proposition 2 show the connection of this question with the surjectivity of the adjoint operator.

Proposition 1. Let (E=\lim\limits_{\to} E_n), (F=\lim\limits_{\to} F_n), let (E) and (F) be separable, and let (E_n) and (F_n) be Fréchet spaces ((n=1,2,3,\ldots)). Let (T) be a continuous linear operator mapping (F) into (E), with sequentially closed image (H). In order that (T) be a weak homomorphism of (F) onto (H), it is necessary and sufficient that (i') be bijective.

Corollary. Under the same assumptions, in order that (T': E'\to F') be surjective, it is necessary and sufficient that (T) be a monomorphism and that (i') be bijective.

Proposition 2. Let (F) be a bornological space, (E) the strict inductive limit of an increasing sequence ((E_n)) of its closed subspaces, and let (T) be a continuous linear operator mapping (F) into (E), with (H=TF). If (T': E'\to F') is surjective, then (i') is bijective.

An inverse spectrum ((X_n)), with maps (\alpha_n), is a sequence
[
\cdots \xrightarrow{\alpha_n} X_n \xrightarrow{\alpha_{n-1}} X_{n-1} \xrightarrow{\alpha_{n-2}} \cdots \xrightarrow{\alpha_2} X_2 \xrightarrow{\alpha_1} X_1
]
of locally convex spaces (X_n) and continuous linear maps (\alpha_n). A morphism of an inverse spectrum ((X_n)) with maps (\alpha_n) into an inverse spectrum ((Y_n)) with maps (\beta_n) is a set of continuous linear maps (\gamma_n: X_n\to Y_n) such that
[
\beta_n\gamma_{n+1}=\gamma_n\alpha_n \qquad (n=1,2,3,\ldots).
]
Inverse spectra and their morphisms form a category. Dually, the category of direct spectra is defined. Consider the functor (\operatorname{Pro}) from the category of inverse spectra to the category of linear spaces, assigning to each inverse spectrum its projective limit (without topology). The functor (\operatorname{Pro}) is left exact; therefore one can speak of its derived functors (\operatorname{Pro}^1,\operatorname{Pro}^2,\ldots). These derived functors were computed in ((^2)).

(I) is called an injective object of the category of locally convex spaces ((TLC)) if a continuous linear mapping into (I) of a subspace of any locally convex space (M) can be extended to a continuous linear mapping of (M) into (I).

Denote by (\mathcal L(\ ,A)) the contravariant functor from the category (TLC) to the category of linear spaces, assigning to each object of (TLC) the linear space of its morphisms into a fixed object (A) of (TLC). Then to the direct spectrum
[
\mathcal E:\quad E_1\to E_2\to\cdots\to E_n\to\cdots
]
there corresponds

inverse spectrum

[
\mathscr L_A(\mathscr E):\ \ldots \to \mathscr L(E_n,A)\to \ldots \to \mathscr L(E_2,A)\to \mathscr L(E_1,A).
]

With the aid of Theorem 11.2 from ((^2)), the following is proved.

Lemma. Let (E=\lim\limits_{\to} E_n) and (\forall n\ \exists p(n)): if (q\ge p(n)), then the topologies induced in (E_n) from (E_{p(n)}) and (E_q) coincide. Then (\operatorname{Pro}^1\mathscr L_I(\mathscr E)=0) for any injective object (I).

Put (G=E/H,\ G_n=E_n/(E_n\cap H)); let (\tau_n) be the quotient topology in (G_n), and let (\tau_{n,p}) be the topology induced in (G_n) from (G_{n+p}). Using the lemma and the fact that every locally convex space is isomorphic to a subspace of an injective object of the category (TLC) ((^2)), it is proved that

Theorem 1. In order that (i:\lim\limits_{\to}(E_n\cap H)\to H) be an isomorphism, it is sufficient that

[
\forall n\ \exists p:\ q\ge p \Rightarrow \tau_{n,p}=\tau_{n,q}.
]

We shall denote by (u_n^0) and (u_n^p) the continuous linear functionals, respectively, on ((G_n,\tau_n)) and on ((G_n,\tau_{n,p})).

Theorem 2. Let (E_n) be normed spaces and suppose the topologies induced in (E_n) from (E_{n+1}) and (E_{n+2}) coincide for all (n). Denote by (B_n) the unit ball in (E_n) and by (\widetilde B_n) the projection of (B_n) in (G_n). For bijectivity of (i') it is necessary and sufficient that

[
\forall n\ \exists p:\ \forall u_n^p\ \forall q\ge p\ \forall \varepsilon>0\ \exists u_n^q:\ |u_n^p(x)-u_n^q(x)|\le \varepsilon\ \forall x\in \widetilde B_n.
]

Proof. Consider the exact sequence of direct spectra
(0\to(E_n\cap H)\to(E_n)\to(G_n)\to0) and pass to the algebraically exact sequence of inverse spectra
(0\to(G_n')\to(E_n')\to((E_n\cap H)')\to0). Denoting the inverse spectra from left to right by (\mathscr L_1,\mathscr L_2,\mathscr L_3), we apply the functor (\operatorname{Pro}). We obtain the canonical exact sequence for a left-exact functor and its derived functors:

[
0\to \operatorname{Pro}\mathscr L_1\to \operatorname{Pro}\mathscr L_2\to \operatorname{Pro}\mathscr L_3\to \operatorname{Pro}^1\mathscr L_1\to \operatorname{Pro}^1\mathscr L_2\to\ldots
]

By the lemma, (\operatorname{Pro}^1\mathscr L_2=0). Between (\operatorname{Pro}\mathscr L_1,\operatorname{Pro}\mathscr L_2,\operatorname{Pro}\mathscr L_3) and, respectively, (G',E',(\lim\limits_{\to}(E_n\cap H))'), there are natural algebraic isomorphisms such that the following diagram is commutative:

[
\begin{array}{ccccccccc}
0&\to&\operatorname{Pro}\mathscr L_1&\to&\operatorname{Pro}\mathscr L_2&\to&\operatorname{Pro}\mathscr L_3&\to&\operatorname{Pro}^1\mathscr L_1\to0\
&&\Vert&&\Vert&&\Vert&&\
0&\to&G'&\longrightarrow&E'&\longrightarrow&(\lim\limits_{\to}(E_n\cap H))'&\to&0\
&&\Vert&&\Vert&&i'\uparrow&&\
0&\to&G'&\longrightarrow&E'&\longrightarrow&H'&\longrightarrow&0
\end{array}
]

Since the lower row is exact, the bijectivity of (i') is equivalent to the equality (\operatorname{Pro}^1\mathscr L_1=0). To complete the proof of the theorem it is enough to introduce in each (G_n') the strong topology and apply Corollary 11.4 from ((^2)).

We shall denote by (\bar u_k) the continuous linear functionals on (E) whose restriction to (E_k\cap H) is zero, and by (\bar v) the continuous linear functionals on (E) whose restriction to (H) is zero.

Corollary. Let (E) be the strict inductive limit of a sequence ((E_n)) of normed subspaces. For bijectivity of (i') it is necessary and sufficient that

[
\forall n\ \exists p:\ \forall \bar u_p\ \forall \varepsilon>0\ \exists \bar v:\ |\bar u_p(x)-\bar v(x)|\le \varepsilon\ \forall x\in B_n.
]

A theorem analogous to Theorem 2 can also be proved in the case where the (E_n) have a countable fundamental system of bounded sets.

In the case where the (E_n) are arbitrary locally convex spaces, the following holds.

Theorem 3. Let (E=\lim\limits_{\longrightarrow} E_n). For the bijectivity of (i') it is sufficient that, for every (n), there exist a neighborhood of zero (U_n) in (E_n) such that (U_n \subset U_{n+1}),

[
\forall n\ \exists p:\ \forall u_n^p\ \forall q \ge p\ \forall \varepsilon > 0\ \exists u_n^q:\quad
|u_n^p(x)-u_n^q(x)| \le \varepsilon\quad \forall x \in \widetilde U_n,
]

where (\widetilde U_n) is the image of (U_n) in (G_n).

Corollary 1. Let (E) be a strict inductive limit of a sequence ((E_n)) of its subspaces. For the bijectivity of (i') it is sufficient that there exist a neighborhood of zero (U) in (E) such that

[
\forall n\ \exists p:\ \forall \bar u_p\ \forall \varepsilon > 0\ \exists \bar v:\quad
|\bar u_p(x)-\bar v(x)| \le \varepsilon\quad \forall x \in E_n \cap U.
]

Corollary 2. Let (E=\lim\limits_{\longrightarrow} E_n), where the mappings of (E_n) into (E_{n+1}) are weakly completely continuous, and the (E_n) are separable ((n=1,2,\ldots)). If (E_n \cap H) is closed in (E_n) for all (n), then (i') is bijective.

In the case where the (E_n) are reflexive Banach spaces, the result of Corollary 2 was obtained by Foiaș and Marinescu ((^{3,4})), and for closed (H), by Komatsu ((^1)).

I express my gratitude to V. P. Palamodov and D. A. Raikov for their attention to the work.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
25 IV 1968

REFERENCES

(^1) H. Komatsu, J. Math. Soc. Japan, 19, No. 3, 366 (1967).
(^2) V. P. Palamodov, Matem. sborn., 75, issue 4, 567 (1968).
(^3) C. Foiaș, G. Marinescu, C. R., 261, No. 23, 4958 (1965).
(^4) C. Foiaș, G. Marinescu, C. R., 263, No. 12, 390 (1966).