MATHEMATICS

V. L. LEVIN

ON NONDEGENERATE SPECTRA OF LOCALLY CONVEX SPACES

(Presented by Academician A. N. Kolmogorov, 21 V 1960)

In the present note two facts are reported that concern nondegenerate spectra of locally convex spaces and are related to certain questions considered in \((^1)\). In addition, the note establishes a necessary and sufficient condition for the perfect completeness (hypercompleteness) of the limit of a nondegenerate inverse spectrum of perfectly complete (hypercomplete) semireflexive spaces, and with the aid of this condition gives a negative answer to one question posed by Kelley.

Definition 1 \((^1)\). An inverse spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) is called standard if, for all \(\alpha<\beta\), \(\pi_\alpha^\beta(X_\beta)\) is dense in \(X_\alpha\), and nondegenerate if the projection \(\pi_\alpha(X)\) of its limit \(X\) is dense in \(X_\alpha\) for every \(\alpha\).

Definition \(1'\) \((^1)\). A direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\) is called standard if all \(\pi_\beta^\alpha\) are one-to-one (so that \(\{Y^\alpha,\pi_\beta^\alpha\}\) may be regarded as a direct spectrum with embeddings), and nondegenerate if its limit \(Y\) is separable.

Definition 2 \((^1)\). The spectrum conjugate to the inverse spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) is the direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\), where \(Y^\alpha=X_\alpha'\) is the strong dual of \(X_\alpha\), and \(\pi_\beta^\alpha=(\pi_\alpha^\beta)^*\) is the mapping conjugate to \(\pi_\alpha^\beta\).

Definition \(2'\) \((^1)\). The spectrum conjugate to the direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\) is the inverse spectrum \(\{\hat X_\alpha,\hat\pi_\alpha^\beta\}\), where \(\hat X_\alpha=(Y^\alpha)'\) is the strong dual of \(Y^\alpha\), and \(\hat\pi_\alpha^\beta=(\pi_\beta^\alpha)^*\) is the mapping conjugate to \(\pi_\beta^\alpha\).

Theorem 1. If the inverse spectrum \(\{\hat X_\alpha,\hat\pi_\alpha^\beta\}\), conjugate to the standard direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\), is nondegenerate, then the spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\) is also nondegenerate.

Theorem 2. The inverse spectrum \(\{\hat X_\alpha,\hat\pi_\alpha^\beta\}\), conjugate to a standard nondegenerate direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\) of semireflexive spaces \(Y^\alpha\), is nondegenerate.

Definition 3 \((^2)\). A separable locally convex space \(X\) is called perfectly complete if, in its dual \(X'\), every weakly closed subspace \(M\) is weakly closed provided it has weakly closed intersections \(M\cap U^{X'}\) with the polars \(U^{X'}\) of all neighborhoods of zero \(U\) of the space \(X\).*

Definition 4 \((^4)\). A separable locally convex space \(X\) is called hypercomplete if, in its dual \(X'\), weakly closed—

* Such spaces, under the name hereditarily \(B\)-complete, were first considered by Pták \((^3)\).

then every circled convex set \(M\), having weakly closed intersections \(M\cap U^{X'}\) with the polars \(U^{X'}\) of all neighborhoods \(U\) of zero in the space \(X\).

Theorem 3. Let \(X\) be the limit of a nondegenerate inverse spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) of complete (hypercomplete) semireflexive spaces \(X_\alpha\). Denote by \(Y\) the limit of the direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\) conjugate to the spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\). In order that the space \(X\) be complete (hypercomplete), it is necessary and sufficient that every subspace (circled convex set) \(M\subset Y\) for which \(\overline{\pi^\alpha}(M)\) is closed in \(Y^\alpha\) for each \(\alpha\), be closed in \(Y\).

Proof. Necessity. Let \(X\) be complete (hypercomplete), and let \(M\) be a subspace (circled convex set) in \(Y\) such that \(\overline{\pi^\alpha}(M)\) is closed in \(Y^\alpha\) for each \(\alpha\). The space conjugate to \(X\) is algebraically isomorphic to \(Y\) \((^5)\). If we prove that the sets \(M\cap U^Y\), where \(U\) runs through a fundamental system of neighborhoods of zero in \(X\), are \(\sigma(Y,X)\)-closed, then from this, by the assumption, it will follow that \(M\) is \(\sigma(Y,X)\)-closed, and hence closed in \(Y\). Thus it suffices to prove that the sets \(M\cap U^Y\) are \(\sigma(Y,X)\)-closed.

Sets of the form \(\overline{\pi_\alpha}^{-1}(U_\alpha)\), where \(U_\alpha\) runs through a fundamental system of neighborhoods of zero in \(X_\alpha\), form a fundamental system of neighborhoods of zero in \(X\). From the nondegeneracy of the spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) there follows the equality
\[ [\overline{\pi_\alpha}^{-1}(U_\alpha)]^Y=\pi^\alpha(U_\alpha^{Y^\alpha}). \]
Further, we have
\[ M\cap U^Y=M\cap[\overline{\pi_\alpha}^{-1}U_\alpha]^Y = M\cap \pi^\alpha(U_\alpha^{Y^\alpha}) \]
and
\[ \overline{\pi^\alpha}(M\cap U^Y) = \overline{\pi^\alpha}(M)\cap \overline{\pi^\alpha}\bigl[\pi^\alpha(U_\alpha^{Y^\alpha})\bigr]. \]
From the nondegeneracy of the inverse spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) its standardness follows \((^1)\), and then all \(\pi^\alpha\) are one-to-one by virtue of the standardness of the direct spectrum \(\{Y^\alpha,\pi_\beta^\alpha\}\), conjugate to the standard inverse spectrum \(\{X_\alpha,\pi_\alpha^\beta\}\) \((^1)\). From the one-to-one nature of \(\pi^\alpha\) follows the equality
\[ \overline{\pi^\alpha}\bigl[\pi^\alpha(U_\alpha^{Y^\alpha})\bigr]=U_\alpha^{Y^\alpha}. \]
Finally we have
\[ \overline{\pi^\alpha}(M\cap U^Y)=\overline{\pi^\alpha}(M)\cap U_\alpha^{Y^\alpha}, \]
whence it is clear that \(\overline{\pi^\alpha}(M\cap U^Y)\) is closed in \(Y^\alpha\), and therefore, since \(\overline{\pi^\alpha}(M\cap U^Y)\) is convex and \(X_\alpha\) is semireflexive, it is \(\sigma(Y^\alpha,X_\alpha)\)-closed. Thus, \(\overline{\pi^\alpha}(M\cap U^Y)\) is \(\sigma(Y^\alpha,X_\alpha)\)-closed and is contained in the \(\sigma(Y^\alpha,X_\alpha)\)-bicompact set \(U_\alpha^{Y^\alpha}\); consequently it is itself \(\sigma(Y^\alpha,X_\alpha)\)-bicompact. Then \(M\cap U^Y\) is \(\sigma(Y,X)\)-bicompact as the image of the \(\sigma(Y^\alpha,X_\alpha)\)-bicompact set \(\overline{\pi^\alpha}(M\cap U^Y)\) under the continuous mapping \(\pi^\alpha\) from the space \((Y^\alpha,\sigma(Y^\alpha,X_\alpha))\) into the space \((Y,\sigma(Y,X))\). Thus, necessity is proved.

Sufficiency. Suppose that every subspace (circled convex set) \(M\subset Y\) for which \(\overline{\pi^\alpha}(M)\) is closed in \(Y^\alpha\) for each \(\alpha\), is closed in \(Y\), and prove that \(X\) is complete (hypercomplete). Let \(M\) be a subspace (circled convex set) in \(Y\), having \(\sigma(Y,X)\)-closed intersections \(M\cap U^Y\) with the polars \(U^Y\) of all neighborhoods \(U\) of zero of the space \(X\). We have
\[ M\cap \pi^\alpha(U_\alpha^{Y^\alpha})=M\cap[\overline{\pi_\alpha}^{-1}(U_\alpha)]^Y, \]
hence \(M\cap \pi^\alpha(U_\alpha^{Y^\alpha})\) is \(\sigma(Y,X)\)-closed. Then, since \(\pi^\alpha\) is one-to-one,
\[ \overline{\pi^\alpha}^{-1}(M)\cap U_\alpha^{Y^\alpha} \]
is \(\sigma(Y^\alpha,X_\alpha)\)-closed as the preimage of the closed set \(M\cap \pi^\alpha(U_\alpha^{Y^\alpha})\) under the continuous mapping \(\pi^\alpha\) from the space \((Y^\alpha,\sigma(Y^\alpha,X_\alpha))\) into the space \((Y,\sigma(Y,X))\). By virtue of the completeness (hypercompleteness) of \(X_\alpha\), from the \(\sigma(Y^\alpha,X_\alpha)\)-closedness of the sets
\[ \overline{\pi^\alpha}^{-1}(M)\cap U_\alpha^{Y^\alpha} \]
it follows that \(\overline{\pi^\alpha}^{-1}(M)\)

is \(\sigma(Y^\alpha, X_\alpha)\)-closed, and hence closed in \(Y^\alpha\). Since this is true for all \(\alpha\), it follows, by assumption, that \(M\) is closed in \(Y\). From the semireflexivity of all \(X_\alpha\) it follows that the conjugate of \(Y\) is algebraically isomorphic to \(X\) \((^5)\), and then from the closedness of the convex set \(M\) it follows that it is \(\sigma(Y, X)\)-closed; but this means that \(X\) is complete (hypercomplete).

Corollary. In order that the strong dual of a reflexive \(LF\)-space \(E\) be complete (hypercomplete), it is necessary and sufficient that in \(E\) every subspace (circled convex set) \(M\) having closed intersections \(M \cap E^n\) with all subspaces of the defining sequence \((E^n)\) be closed.

Proof. In the dual \(E'\) of the \(LF\)-space \(E\), the strong topology coincides with the topology of the limit of the inverse spectrum \(\{E_n,\varphi_n^{\,m}\}\) dual to the direct spectrum \(\{E^n\}\) with embeddings. It is easy to see that the spectrum \(\{E_n,\varphi_n^{\,m}\}\) is nondegenerate. Since the reflexivity of \(E\) implies the reflexivity of all \(E^n\), their strong duals \(E_n\) are hypercomplete as reflexive \(DF\)-spaces\(^*\), and the assertion of the corollary is obtained from Theorem 3, applied to the limit of the inverse spectrum \(\{E_n,\varphi_n^{\,m}\}\).

Remark. As D. A. Raikov observed, the corollary proved above can also be formulated as follows: in order that the strong dual of a reflexive \(LF\)-space \(E\) be complete (hypercomplete), it is necessary and sufficient that in \(E\) every subspace (circled convex set) \(M\) containing the limits of all its convergent sequences be closed.

Kelley in \((^4)\) expressed the supposition that the product and the direct sum of a countable family of hypercomplete spaces are hypercomplete. We shall show that the product of a sequence of complete (hypercomplete) spaces may fail to be a complete (hypercomplete) space.

Let \(E\) be a reflexive \(LF\)-space and let there exist in \(E\) a nonclosed subspace \(M\) having closed intersections \(M \cap E^n\) with all spaces of the defining sequence \((E^n)\)\(^{**}\). Consider the inverse spectrum \(\{E_n,\varphi_n^{\,m}\}\) dual to the spectrum with embeddings \(\{E^n\}\).

All \(E_n\) are hypercomplete (and hence complete) as reflexive \(DF\)-spaces. The limit of the spectrum \(\{E_n,\varphi_n^{\,m}\}\) is closed in the product of the spaces \(E_n\) and, being isomorphic to the strong dual of the space \(E\), is not complete by virtue of the corollary to Theorem 3. Therefore, since closed subspaces of complete spaces are complete \((^2)\), the product of the spaces \(E_n\) is not complete (and hence not hypercomplete).

Remark. As D. A. Raikov noted in his review of Collins’s paper \((^2)\) (see \((^8)\)), the direct sum of a sequence of complete (hypercomplete) spaces may fail to be a complete (hypercomplete) space.

Indeed, let \(E\) be an incomplete \(LF\)-space\(^{***}\), \((E^n)\) its defining sequence. The spaces \(E^n\) are hypercomplete (and hence complete) as \(F\)-spaces. From the separability of \(E\) it follows that it is isomorphic to the quotient space, by a closed subspace, of the direct sum of the spaces \(E^n\). Since quotient spaces by closed subspaces of complete spaces

\(^*\) The proof of hypercompleteness of a reflexive \(DF\)-space can be carried out according to the plan of the proof of Theorem 6.5 in \((^6)\).

\(^ {**}\) Grothendieck showed (see \((^7)\), § 2) that such spaces exist.

\(^ {***}\) As follows from one example of Grothendieck (see \((^7)\), § 2), such spaces exist.

are perfectly complete \((^{2,3})\), while \(E\) is not perfectly complete, then the direct sum of the spaces \(E^n\) is not perfectly complete (and hence is not hypercomplete either).

Moscow State University
named after M. V. Lomonosov

Received
17 V 1960

References

\(^{1}\) D. A. Raikov, Tr. Moskovsk. matem. obshch., 7, 413 (1958).
\(^{2}\) H. S. Collins, Trans. Am. Math. Soc., 79, 256 (1955).
\(^{3}\) V. Pták, Czechoslov. matem. zhurn., 3, 78 (1953).
\(^{4}\) J. L. Kelley, The Michigan Math. J., 5, No. 2 (1958).
\(^{5}\) O. Takenouchi, Math. J. Okayama Univ., 2, 57 (1952).
\(^{6}\) V. Pták, Bull. Soc. Math. de France, 86, 41 (1958).
\(^{7}\) A. Grothendieck, Summa Brasiliensis Math., 3 (1954) (Russian transl. Matematika, 2, 3 (1958)).
\(^{8}\) RZhMatem., Abstract 8130 (1956).