B. Z. Vulikh

On Linear Lattices Equivalent to Lattices with a Monotone Norm

(Presented by Academician V. I. Smirnov on 31 V 1962)

Unlike the terminology adopted in [1], we shall call a normed (respectively Banach) lattice any linear lattice that is at the same time also a normed (respectively Banach) space. In doing so, no requirement is imposed that the ordering and the norm be in any way coordinated with one another. In particular, a normed lattice need not be Archimedean. We shall say that the norm in a normed lattice is monotone if the inequality \(|x| \le |y|\) implies \(\|x\| \le \|y\|\). A lattice with a monotone norm is called a \(KN\)-lineal [1]. A \(KN\)-lineal that is complete with respect to \((b)\)-convergence, i.e., convergence in norm, is called a \(KB\)-lineal*. Every \(KN\)-lineal is an Archimedean lattice.

Definition. We shall call a normed lattice \(X\) equivalent to a \(KN\)-lineal if the norm given in \(X\) is equivalent to some monotone norm.

In [1] a condition is given that is necessary and sufficient for a normed lattice to be equivalent to a \(KN\)-lineal. It is easy to see that the formulation introduced there is equivalent to the following:

There exists a constant \(M\) such that from \(|x| \le |y|\) there follows the inequality \(\|x\| \le M\|y\|\).

In this note, a number of further criteria are established for the equivalence of a given normed lattice to a \(KN\)-lineal of one type or another. In the formulations of these criteria the following concepts are used, introduced by M. G. Krein and M. A. Krasnosel’skii in the general theory of cones in normed spaces: the cone of positive elements \(X_+\) in a partially ordered normed space \(X\) is called normal [2] if there exists a \(\delta > 0\) such that \(\|x+y\| \ge \delta\) for all \(x,y \in X_+\) with norms \(\|x\|=\|y\|=1\); the cone \(X_+\) is called regular (respectively fully regular**) [4] if every increasing sequence of elements of the space \(X\) that is bounded with respect to the ordering (respectively with respect to the norm) \((b)\)-converges to some limit. A fully regular cone \(X_+\) is also regular.

Theorem 1. In order that a Banach lattice \(X\) be equivalent to a \(KB\)-lineal, it is necessary and sufficient that the cone \(X_+\) be closed and normal.

* A linear lattice \(X\) is called Archimedean if, from the validity for some \(x,y \in X\) and any natural \(n\) of the inequality \(nx \le y\), it follows that \(x \le 0\).

** I. A. Bakhtin proved [3] that for normality of the cone \(X_+\) it is necessary and sufficient that there exist a constant \(M\) such that from \(0 \le y \le x\) there follows the inequality \(\|y\| \le M\|x\|\). This criterion is the most convenient for checking normality of the cone \(X_+\) in concrete spaces. The proof of I. A. Bakhtin’s criterion is contained implicitly in [1] (p. 393).

Proof. The necessity of these conditions is easily verified for a monotone norm, i.e., in any \(KN\)-lineal, and then these conditions must also be satisfied for a norm equivalent to a monotone one.

Sufficiency. By a theorem of M. G. Krein—V. L. Shmul’yan \((^{5,6})\), for every \(x \in X\) there exists a \(u \geq x_+\) such that \(\|u\| \leq M_1\|x\|\), where \(M_1\) is a constant independent of \(x\). With the help of this inequality it is easy to obtain that, if \(M\) is the constant from the normality condition for the cone \(X_+\), given by I. A. Bakhtin, then \(\||x|\| \leq 2MM_1\|x\|\), and then from the inequality \(0 \leq y \leq |x|\) it will follow that

\[ \|y\| \leq M\|x\| \leq 2M^2M_1\|x\|. \]

Now we introduce a new norm by setting

\[ \||x|\|^*=\sup_{0\leq y\leq |x|}\|y\|. \]

This norm will be monotone and equivalent to the original norm \(\|x\|\).

In Theorem 1 one cannot dispense with the requirement that the cone \(X_+\) be closed, replacing it by the weaker condition: the structure \(X\) is Archimedean. It is known that there exist Banach Archimedean structures with a normal but nonclosed cone*.

It turns out that in an arbitrary normed structure \(X\), closedness and normality of the cone \(X_+\) are no longer sufficient for the possibility of introducing an equivalent monotone norm. This is confirmed by the following example.

Let \(X\) consist of all functions of bounded variation \(x(t)\) on \([0,1]\); the positive elements in \(X\) are all nondecreasing functions \(x(t)\) for which \(x(0)\geq 0\); the norm is introduced by the formula \(\|x\|=\sup |x(t)|\). It is easy to see that in this space the cone \(X_+\) is closed and normal. At the same time it is not difficult to find elements \(x\in X\) for which \(\||x|\|\) is arbitrarily large in comparison with \(\|x\|\). Consequently, there exists a sequence \(x_n \xrightarrow{(b)} 0\) such that \(\||x_n|\| \nrightarrow 0\). But in a \(KN\)-lineal this is impossible, and therefore an equivalent monotone norm in \(X\) does not exist.

We shall say that the structural operations in a normed structure \(X\) are \((b)\)-continuous if from \(x_n \xrightarrow{(b)} x\) and \(y_n \xrightarrow{(b)} y\) it follows that \(x_n \vee y_n \xrightarrow{(b)} x \vee y\) (an equivalent condition: from \(x_n \xrightarrow{(b)} x\) it follows that \(x_n^+ \xrightarrow{(b)} x_+\)). From the \((b)\)-continuity of the structural operations follows the closedness of the cone \(X_+\).

Theorem 2. In order that a normed structure \(X\) be equivalent to a \(KN\)-lineal, it is necessary and sufficient that the cone \(X_+\) be normal and that the structural operations in \(X\) be \((b)\)-continuous*.

Proof. The necessity of the conditions follows from the same considerations as in Theorem 1. Conversely, from the \((b)\)-continuity of the structural operations it easily follows that there is a constant \(M_1\) such that \(\|x_+\|\leq M_1\|x\|\). The rest of the argument is the same as in the proof of Theorem 1.

In any Archimedean \(K\)-lineal the notion of \((r)\)-convergence (convergence with a regulator) makes sense: \(x_n \xrightarrow{(r)} x\) means that \(|x_n-x|\leq \varepsilon_n y\), where \(\varepsilon_n \to 0\). We shall say that \(x_n \xrightarrow{(*)r} x\) if from any subsequence \(\{x_{n_i}\}\) one can extract a subsequence \(x_{n_{i_k}} \xrightarrow{(r)} x\).

* The Archimedean principle follows from the closedness of the cone \(X_+\) in any partially ordered normed space.

** A simple example of such a structure was constructed by G. Ya. Lozanovskii by means of a certain reordering of the classical space \(C\).

*** This theorem can be obtained from a more general theorem concerning linear topological structures (see \((^7)\), Theorem 8.1).

Theorem 3. In order that an Archimedean Banach structure \(X\) be equivalent to a \(KB\)-lineal, it is necessary and sufficient that \((b)\)-convergence in \(X\) coincide with \((*)_r\)-convergence.

Proof. The necessity of the condition follows from Birkhoff’s theorem, according to which in every \(KB\)-lineal \((b)\)-convergence coincides with \((*)_r\)-convergence \((^1)\). Conversely, the condition of the theorem immediately implies the closedness of the cone \(X_+\), since in every linear structure it is closed with respect to \((r)\)-convergence. It remains to verify the normality of \(X_+\). Suppose that the cone \(X_+\) is not normal. Then we find two sequences of elements \(\{x_n\}\) and \(\{y_n\}\) such that \(0 \leq y_n \leq x_n\), \(x_n \xrightarrow{(b)} 0\), and \(y_n \not\xrightarrow{(b)} 0\). Further, from any subsequence \(\{x_{n_i}\}\) one can select a subsequence \(x_{n_{i_k}} \xrightarrow{(r)} 0\). Then also \(y_{n_{i_k}} \xrightarrow{(r)} 0\), and therefore \(y \xrightarrow{(*)_r} 0\). We obtain a contradiction in view of the coincidence of \((*)_r\)-convergence with \((b)\)-convergence.

In an arbitrary \(KN\)-lineal Birkhoff’s theorem is not true. An example may be the subspace of the space \(\mathcal L\) consisting of all bounded functions, with the usual ordering. Therefore Theorem 3 does not carry over in full to arbitrary normed structures, and only the following is valid.

Theorem 4. If in a normed structure \(X\) \((b)\)-convergence coincides with \((*)_r\)-convergence, then \(X\) is equivalent to a \(KN\)-lineal.

For the proof, instead of the closedness of the cone \(X_+\), one must verify the \((b)\)-continuity of the structural operations in \(X\). But the latter is a simple consequence of their \((r)\)-continuity.

Let us now turn to the characterization of normed structures equivalent to a \(KB\)-space. It is easy to see that the definition of a \(KB\)-space given by L. V. Kantorovich (see, for example, \((^1)\)) is equivalent to the following: a \(KB\)-space is a \(K_{\sigma}N\)-space \(X\) with a regular cone \(X_+\), in which the norm is monotonically complete. The latter means that if \(x_n \geq 0\) and \(x_n \uparrow +\infty\), then \(\|x_n\|\to\infty\). Equivalently, a \(KB\)-space can be defined as a \(KN\)-lineal \(X\) with a fully regular cone \(X_+\).

Indeed, the cone \(X_+\) in a \(KN\)-lineal \(X\) is closed, and from its closedness and regularity it follows that \(X\) is a \(K_{\sigma}N\)-space. Moreover, from the full regularity and closedness of the cone \(X_+\) follows the monotone completeness of the norm. The converse is obvious: in every \(KB\)-space \(X\) the cone \(X_+\) is fully regular.

Theorem 5. In order that a normed structure \(X\) be equivalent to a \(KB\)-space, it is necessary and sufficient that the cone \(X_+\) be fully regular, and that the structural operations in \(X\) be \((b)\)-continuous.

Proof. The necessity of the conditions is obvious, since the property of full regularity of the cone \(X_+\) is preserved under passage to an equivalent norm. Conversely, from the full regularity of the cone \(X_+\) follows its normality \((^4)\), and then, by Theorem 2, the structure \(X\) is equivalent to a \(KN\)-lineal. But since the cone of positive elements in this \(KN\)-lineal will also be fully regular, this \(KN\)-lineal will be a \(KB\)-space.

We note that the requirement of \((b)\)-continuity of the structural operations in the condition of Theorem 5 may be replaced by the following, weaker one: from \(x_n \xrightarrow{(b)} 0\) it follows that \(|x_n| \xrightarrow{(b)} 0\). This requirement is equivalent to the \((b)\)-continuity of the structural operations in the case when the cone \(X_+\) is normal. The same remark is applicable also to Theorem 2.

Since a \(KB\)-space is always Banach, we thereby conclude that from the conditions of Theorem 5 follows the \((b)\)-completeness of the structure \(X\). For a seminormed additive structure \(X\), where the cone \(X_+\) is fully regular and the requirement of \((b)\)-continuity of the structural operations is replaced by another, more restrictive one \((\|x\|=\||x|\|)\), the \((b)\)-completeness of \(X\) was established in \((^8)\).

We note that in the \(K\)-space of functions of bounded variation considered in connection with Theorem 1, the cone \(X_+\) is quite regular and closed. However, this space is not equivalent to a \(KB\)-space, since an equivalent monotone norm cannot be introduced in it. At the same time, if \(X\) is a Banach structure, then it is clear that in the condition of Theorem 5 the requirement \((b)\)—continuity of the structural operations—can be replaced by the closedness of the cone \(X_+\). The condition of complete regularity of the cone alone will again be insufficient here.

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
29 V 1962

REFERENCES

1. B. Z. Vulikh, Introduction to the theory of partially ordered spaces, 1961.
2. M. G. Kreĭn, DAN, 28, No. 1, 13 (1940).
3. I. A. Bakhtin, Dissertation, Voronezh State Univ., 1958.
4. M. A. Krasnosel’skiĭ, DAN, 135, No. 2, 255 (1960).
5. I. A. Bakhtin, M. A. Krasnosel’skiĭ, V. Ya. Stetsenko, Sibirsk. matem. zhurn., 3, No. 1, 156 (1962).
6. M. G. Kreĭn, DAN, 28, No. 1, 18 (1940).
7. J. Namioka, Mem. Am. Math. Soc., No. 24 (1957).
8. M. Zamansky, Bull. Soc. Math. Belg., 10, No. 2, 67 (1958).