UDC 513.88 : 517.4./5

MATHEMATICS

I. P. KOSTENKO

ON THE REFLEXIVITY OF SOME COMBINATIONS OF PARTIALLY ORDERED SPACES

(Presented by Academician L. V. Kantorovich on 25 I 1967)

1°. In the theory of \(K\)-spaces, an essential role is played by an operation, introduced by A. G. Pinsker, which makes it possible to construct, from a given family of \(K\)-spaces \(X_\xi\) \((\xi \in \Xi)\), numerous new \(K\)-spaces. This is the operation of combination, generalizing the direct-sum operation known in algebra. It is multivalued. The combination uniquely defined in \((^1)\) we call complete, in accordance with the terminology of \((^2)\)*. It is the largest of all possible combinations; the others are normal subspaces of the complete one containing all \(X_\xi\) \((\xi \in \Xi)\). Notation: \(S_{\xi \in \Xi} X_\xi\).

The smallest of the possible combinations we call finite—this is the direct sum of the spaces with coordinatewise order. In other words, the combination \(X = SX_\xi\) is called finite if, for any \(x \in X\), \(x = Sx_\xi\) \((x_\xi \in X_\xi)\), \(x_\xi \ne 0\) only for a finite number of indices \(\xi\).

In this note the following problem is posed: given a family of reflexive (with respect to order) \(K\)-spaces \(X_\xi\) \((\xi \in \Xi)\); which combinations of these spaces are reflexive, and which are not? A particular case of combinations is that of discrete \(K\)-spaces, considered as combinations of one-dimensional spaces; therefore the question posed also contains another: what are the conditions for reflexivity of a discrete \(K\)-space? This question intersects with an analogous question from the theory of Köthe sequence spaces: which spaces are perfect (see \((^3)\))? The concept of a reflexive \(K\)-space is close to the concept of a Köthe space introduced in \((^4)\).

Reflexivity (with respect to order) of a \(K\)-space \(X\) was first considered by H. Nakano (see \((^5)\)). He showed that if in \(X\) there exists a sufficient set of fully linear functionals, then under the canonical embedding of \(X\) into \(\overline{X}^{**}\) by means of the correspondence \(x \to F_x\), \(F_x(f) = f(x)\), \(x \in X\), \(f \in \overline{X}\), \(X\) turns out to be a normal subspace in \(\overline{X}\). If all of \(\overline{X}\) is exhausted by functionals of the form \(F_x\), then the \(K\)-space \(X\) is called reflexive***. H. Nakano also established a certain general criterion of reflexivity (see \((^1)\), p. 290).

Examples of reflexive combinations are the \(K\)-spaces \(l^p\) \((p \ge 1)\), \(m\), and \(l[\Xi]\)—the space of functions nonzero at no more than a countable set of points of \(\Xi\) and such that

\[ \sum_{\xi \in \Xi} |x(\xi)| < +\infty. \]

Examples of nonreflexive combinations are the space \(c_0\) of sequences converging to zero, and \(m[\Xi]\) \((\Xi\) uncountable)—the \(K\)-space—

* The elements of the complete combination are all possible families \(\{x_\xi\}_{\xi \in \Xi}\), \(x_\xi \in X_\xi\). For example, the space of all real sequences \(s\) is the complete combination of one-dimensional spaces. In what follows we use, without special explanation, the terminology from \((^1)\).

** \(\overline{X}\) is the \(K\)-space of fully linear functionals on \(X\).

*** Reflexivity with respect to order differs from reflexivity with respect to the norm. Thus, for the \(KN\)-space \(m\), the order dual is \(l\), and it does not coincide with the norm dual.

of functions bounded with respect to \(\Xi\) and different from zero at no more than a countable set of points.

\(2^\circ.\) It is quite easy to show that complete and finite sums of reflexive spaces are reflexive. With the aid of Theorems IX.4.6 and IX.4.7 from [1] one can prove that the \(K N\)-space of bounded elements with a sufficient set of completely linear functionals is reflexive. It is interesting that such a space is not norm-reflexive, except in the finite-dimensional case.

Definition 1. Let \(X = S_{\xi\in\Xi} X_\xi\) and \(\Xi' \subset \Xi\). The set \(X'\) of all elements \(x \in X\) of the form \(x = Sx_\xi\), where \(x_\xi = 0\) for \(\xi \in \overline{\Xi'}\), forms a component in \(X\). We shall call it a subsum of the sum \(X\). Notation: \(X' = S_{\xi\in\Xi'} X_\xi\).

Definition 2. We shall say that the sum \(X = S_{\xi\in\Xi} X_\xi\) has property \((R_0)\) if in it one cannot choose a system of elements \(\{y_\xi\}_{\xi\in\Xi}\) such that: 1) \(y_\xi \in X_\xi^+\); 2) the system is unbounded, and 3) in every reflexive subsum \(X' = S_{\xi\in\Xi'} X_\xi\) there exists \(y' = S_{\xi\in\Xi'} y_\xi\).

The space \(l[0,1]\) has this property, while \(m[0,1]\) does not.

Theorem 1. In order that the sum \(X = SX_\xi\), where all \(X_\xi\) are reflexive, be reflexive, it is necessary and sufficient that it have property \((R_0)\).

This theorem helps to distinguish two types of sums for which the question of reflexivity is solved more simply.

Definition 3. Let \(X = S_{\xi\in\Xi} X_\xi\), and let \(\{\Xi_\beta\}_{\beta\in B}\) be the collection of all countable subsets of \(\Xi\). If every subsum \(X^{(\beta)} = S_{\xi\in\Xi_\beta} X_\xi\) \((\beta \in B)\) is reflexive, we shall call \(X\) a countably reflexive sum.

Definition 4. We shall say that the sum \(X = S_{\xi\in\Xi} X_\xi\) has property \((R_1)\) if in it one cannot choose a system \(\{y_\xi\}_{\xi\in\Xi}\) such that: 1) \(y_\xi \in X_\xi^+\); 2) the system is unbounded, and 3) in every subsum \(X' = S_{\xi\in\Xi'} X_\xi\), where \(\Xi'\) is countable, there is an element \(y' = S_{\xi\in\Xi'} y_\xi\).

Let us note that if \(X\) is a countably reflexive sum, then formally property \((R_1)\) is stronger for it than property \((R_0)\). The space \(l[0,1]\) has property \((R_1)\), while \(m[0,1]\) does not.

Theorem 2. In order that a countably reflexive sum \(X = SX_\xi\) be reflexive, it is necessary and sufficient that it have property \((R_1)\).

Thus, if \(X = |SX_\xi|\) is not reflexive, then either it is countably reflexive, and then Theorem 2 indicates the reason for the failure of its reflexivity, or it is not countably reflexive, and then some subsum \(X' = S_{\xi\in\Xi'} X_\xi\), \(\Xi'\) countable, is not reflexive. Hence one may restrict oneself to the investigation of sums of a countable set of reflexive spaces. We shall formulate this theorem in other terms.

Theorem \(2'\). In order that a countably reflexive sum \(X = S_{\xi\in\Xi} X_\xi\) be reflexive, it is necessary and sufficient that it contain no linear substructure, isomorphic (linearly and structurally) to the space \(m[\Xi']\) (\(\Xi'\) uncountable) and situated in \(X\) so that, if \(y_\xi\) \((\xi \in \Xi')\) are the elements of \(X\) corresponding under this isomorphism to the unit elements \(e_\xi \in m[\Xi']\), then: 1) \(y_\xi \in X_\xi^+\) and 2) \(\{y_\xi\}_{\xi\in\Xi'}\) is unbounded.

In the necessity part, the proofs of both theorems go through for arbitrary sums.

\(3^\circ.\) Definition 5. Let \(Y\) be a \(K\)-space, and let \(\{X_\alpha\}_{\alpha\in A}\) be some system of its normal subspaces. The collection \(X\) of all such

elements \(x \in Y\) which are representable in the form

\[ x=\sum_{i=1}^{k} x_{\alpha_i}, \quad x_{\alpha_i}\in X_{\alpha_i} \ * \]

we shall call the sum of the spaces \(X_\alpha\) \((\alpha \in A)\). Notation:

\[ X=\sum_{\alpha\in A} X_\alpha . \]

The sum is also a \(K\)-space.

Definition 6. Let

\[ X=\underset{\xi\in \Xi}{S} X_\xi . \]

If there exists a system of subsets \(\{\Xi_\beta\}_{\beta\in B}\) of the set \(\Xi\) such that each subconnection

\[ X^{(\beta)}=\underset{\xi\in \Xi_\beta}{S} X_\xi \]

is reflexive and

\[ X=\sum_{\beta\in B} X^{(\beta)}, \]

we shall call \(X\) a quasi-reflexive connection of the spaces \(X_\xi\) \((\xi\in \Xi)\), and the system \(\{\Xi_\beta\}_{\beta\in B}\) the carrier of this connection.

Let us note that a single quasi-reflexive connection may have many different carriers, but we assume that some carrier has been chosen and fixed. Any subconnection

\[ X'=\underset{\xi\in \Xi'}{S} X_\xi \]

of the quasi-reflexive connection

\[ X=\underset{\xi\in \Xi}{S} X_\xi \]

with carrier \(\{\Xi_\beta\}_{\beta\in B}\) may be considered as an independent quasi-reflexive connection with carrier \(\{\Xi_\beta\cap \Xi'\}_{\beta\in B}\). The question of the reflexivity of these connections is completely settled by Theorems 3, 4, and 5.

Definition 7. We shall call the carrier \(\{\Xi_\beta\}_{\beta\in B}\) of the quasi-reflexive connection

\[ X=\underset{\xi\in \Xi}{S} X_\xi \]

saturated if: 1) among the \(\Xi_\beta\) \((\beta\in B)\) there is no finite number covering \(\Xi\), and 2) whatever infinite \(\Xi'\subset \Xi\) is taken, there exists \(\Xi_\beta\) such that \(\Xi'\cap \Xi_\beta\) is infinite.

Theorem 3. If, in the connection

\[ X=\underset{\xi\in \Xi}{S} X_\xi \]

with carrier \(\{\Xi_\beta\}_{\beta\in B}\), every subconnection

\[ X'=\underset{\xi\in \Xi'}{S} X_\xi, \]

whose carrier is saturated, is reflexive, then \(X\) itself is reflexive.

Corollary. For the reflexivity of the connection

\[ X=\underset{\xi\in \Xi}{S} X_\xi \]

with carrier \(\{\Xi_\beta\}_{\beta\in B}\), it is sufficient that it contain no subconnection

\[ X'=\underset{\xi\in \Xi'}{S} X_\xi, \]

whose carrier \(\{\Xi_\beta\cap \Xi'\}_{\beta\in B}\) is saturated.

Let us note that the condition of the corollary concerns only the carrier. A trivial example of a carrier possessing the indicated property is furnished by the system of all finite subsets of an infinite set.

Definition 8. Let

\[ X=\underset{\xi\in \Xi}{S} X_\xi \]

be a quasi-reflexive connection with carrier \(\{\Xi_\beta\}_{\beta\in B}\). We shall say that it has property \((R_2)\), if in it one cannot choose a system \(\{y\}_{\xi\in \Xi}\) such that: 1) \(y_\xi\in X_\xi^{+}\); 2) the system is unbounded, and 3) in each subconnection

\[ X^{(\beta)}=\underset{\xi\in \Xi_\beta}{S} X_\xi \]

there is an element

\[ y^{(\beta)}=\underset{\xi\in \Xi_\beta}{S} y_\xi . \]

Theorem 4. In order that a quasi-reflexive connection

\[ X=\underset{n\in N}{S} X_n \]

(\(N\) is the natural series), whose carrier \(\{N_\beta\}_{\beta\in B}\) is saturated, be reflexive, it is necessary and sufficient that it have property \((R_2)\).

Examples. Consider all possible systems of almost disjoint*** subsets of \(N\). Introduce in this collection the ordering by inclusion. Zorn’s lemma guarantees the existence of maximal systems—

* This representation is unique for each \(x\in X\) only in the case when the \(X_\alpha\) \((\alpha\in A)\) are pairwise disjoint. In this case the sum becomes a finite connection.

** A finite connection

\[ X=\underset{n\in N}{S} X_n \]

with carrier \(\{n\}_{n\in N}\) does not have this property, whereas with carrier \(\{N\}\) it does.

*** \(N'\) is almost disjoint from \(N''\) if \(N'\cap N''\) is finite or empty.

system, while among them there are saturated ones. Let \(T=\{N_\beta\}_{\beta\in B}\) be such. Consider \(m\) as a union of one-dimensional \(X_n\) \((n\in N)\) and single out the system of subunions

\[ m^{(\beta)}=\sum_{n\in N_\beta} X_n\quad(\beta\in B). \]

The quasireflexive union

\[ m[\dot T]=\sum_{\beta\in B} m^{(\beta)} \]

does not possess property \((\mathrm{R}_2)\), and therefore is not reflexive.

Similarly one constructs \(s[\dot T]\), \(l[\dot T]\), etc. An example of a carrier not satisfying the condition in the corollary to Theorem 3 is provided by the system \(\{N_x\}_{x\in(0,1)}\) of almost disjoint sets, defined by Sierpiński in \((^6)\), supplemented by the set

\[ N_0=N\setminus\bigcup_{x\in(0,1)} N_x. \]

Denote it by \(T^{(S)}\) and, as above, construct the reflexive spaces \(s[T^{(S)}]\), \(m[T^{(S)}]\), and \(l[T^{(S)}]\). These spaces will also be perfect sequence spaces of Köthe. The space \(\delta\), introduced in \((^7)\), is a quasireflexive union with a saturated carrier, not possessing property \((\mathrm{R}_2)\) (see also \((^3)\)).

The presence of property \((\mathrm{R}_2)\) in the quasireflexive union \(X=SX_n\) can be connected with the space \(m[\dot T]\).

Theorem 4. In order that the quasireflexive union

\[ X=\sum_{n\in N} X_n \]

be reflexive, it is necessary and sufficient that it contain no linear substructure isomorphic to the space \(m[\dot T]\) and situated in \(X\) in such a way that, if \(y_{n_k}\) \((k=1,2,\ldots)\) are the elements of \(X\) corresponding under this isomorphism to the vectors \(e_k\in m[\dot T]\), then: 1) \(y_{n_k}\in X_{n_k}+\) \((n_k\) are distinct for \(k=1,2,\ldots)\); 2) the system \(\{\lambda_k y_{n_k}\}_{k=1,2,\ldots}\) is unbounded for arbitrary numbers \(\lambda_k\ne0\).

This theorem also contains one more necessary criterion for the reflexivity of an arbitrary union, since in proving the necessity of the condition of the theorem the quasireflexivity of \(X\) is not used. From Theorems 2, 3, and 4 it follows easily that

Theorem 5. In order that the quasireflexive union

\[ X=\sum_{\xi\in\Xi} X_\xi \]

with carrier \(\{\Xi_\beta\}_{\beta\in B}\) be reflexive, it is necessary and sufficient that: 1) \(X\) possess property \((\mathrm{R}_1)\), and 2) every subunion

\[ X'=\sum_{\xi\in\Xi'} X_\xi, \]

for which \(\Xi'\) is countable and the carrier is saturated, possess property \((\mathrm{R}_2)\).

4°. Let us formulate a third necessary criterion for the reflexivity of an arbitrary union.

Definition 9. We shall say that the union

\[ X=\sum_{\xi\in\Xi} X_\xi \]

possesses property \((\mathrm{R}_3)\) if in it one cannot choose a system \(\{y_n\}_{n\in N}\) such that: 1) \(y_n\in X_{\xi_n}\), the \(\xi_n\) being pairwise distinct; 2) the system is unbounded, and 3) for every sequence of numbers \(\lambda_n\to0\), the element

\[ \sum_{n\in N}\lambda_n y_n \]

is contained in \(X\).

Analogously to the preceding, this property is connected with the absence in \(X\) of a substructure linearly and structurally isomorphic to the space \(c_0\) and situated in \(X\) in a certain way.

Theorem 6. In order that the union \(X=SX_\xi\) be reflexive, it is necessary that it possess property \((\mathrm{R}_3)\).

From the three criteria noted above one easily obtains three necessary criteria for the reflexivity of an arbitrary \(K\)-space.

The author expresses gratitude to his scientific adviser, Prof. B. Z. Vulikh, for his attention to the present work.

Leningrad Pedagogical Institute
named after A. I. Herzen

Received
21 I 1967

CITED LITERATURE

1. B. Z. Vulikh, Introduction to the Theory of Semi-Ordered Spaces, 1961.
2. L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semi-Ordered Spaces, 1950.
3. R. Cook, Infinite Matrices and Sequence Spaces, 1960.
4. J. Dieudonné, J. Analyse Math., 1, 81 (1951).
5. H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo, 1950.
6. W. Sierpiński, Cardinal and Ordinal Numbers, Warszawa, 1958.
7. G. Köthe, O. Toeplitz, J. reine u. angew. Math., 171, 193 (1934).