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MATHEMATICS
G. Ya. Lozanovskii
ON TOPOLOGICALLY REFLEXIVE \(KB\)-SPACES
(Presented by Academician V. I. Smirnov, 18 IV 1964)
It is known that many other Banach spaces which turn out to be reflexive* are closely connected with the space \(L\) of summable functions; for example, \(L_p\) for \(p>1\) and many Orlicz spaces. Recall that an Orlicz space, in the case when the functions are given on a bounded closed subset of Euclidean space, is reflexive if and only if the defining \(N\)-function \(M(u)\) and the complementary \(N\)-function satisfy the \(\Delta_2\)-condition \(\left((^{1}),\text{ p. }152\right)\).
In this note it is shown that the theory of semi-ordered spaces makes it possible to construct analogous, and even more varied, reflexive and at the same time \(KB\)-spaces, starting from an arbitrary \(KB\)-space. The notion of a \(KB\)-space, as is known, was introduced by L. V. Kantorovich. It is a \(K\)-space in which a norm is defined that turns it into a Banach space, and the following conditions are satisfied: 1) from \(|x|\le |y|\) it follows that \(\|x\|\le \|y\|\) (monotonicity of the norm); 2) if \(x_n\downarrow 0\), then \(\|x_n\|\to 0\); 3) if \(x_n\uparrow +\infty\), then \(\|x_n\|\to +\infty\) (see also \((^{2})\)).
Let \(X\) be an arbitrary \(KB\)-space in which a complete set of pairwise disjoint positive elements has been distinguished; \(\hat X\) is the maximal extension of the space \(X\) \((^{2})\). It makes sense \((^{3})\) to speak of continuous functions defined on \(\hat X\). Put, for arbitrary \(p\ge 1\),
\[ X_p=\{x:x\in\hat X,\ |x|^p\in X\}, \]
i.e. \(X_p\) consists of all elements of \(\hat X\) whose \(p\)-th power of the modulus is contained in \(X\). We introduce on \(X_p\) a norm by setting
\[ \|x\|_p=\||x|^p\|^{1/p}, \]
where \(\|\ \ \|\) is the norm in the original \(KB\)-space \(X\).
Theorem 1. \(X_p\) for \(p>1\) is a reflexive \(KB\)-space.
We note that if \(X=L\), then \(X_p\) coincides with the usual space \(L_p\). Although Theorem 1 is a direct generalization of the known theorem on the reflexivity of the spaces \(L_p\), nevertheless the proof of Theorem 1 is carried out by an entirely different method and relies on certain considerations from the theory of semi-ordered spaces and on D. P. Milman’s theorem on the reflexivity of uniformly convex spaces (it is not the spaces \(X_p\) themselves that turn out to be uniformly convex, but certain auxiliary spaces).
Let us also dwell on the spaces \(L_{(\omega)}\), considered by Halperin, for example, in \((^{4})\). If we put \(X=L_{(\omega)}^1\) and if this \(X\) is a \(KB\)-space, then \(X_p=L_{(\omega)}^p\) for \(p>1\) is a reflexive \(KB\)-space.
The spaces \(L^P\), where \(P=(p_1,p_2,\ldots,p_n)\), considered in detail by A. Benedek and Panzone \((^{5})\), are also closely connected with the spaces \(X_p\) introduced by us. If \(1<p_i<\infty\), where \(i=1,2,\ldots,n\), then \(L^P\) coincides with our space \(X_p\), constructed from \(X=L^Q\), where \(p=\min p_i\) and \(Q=(p_1/p,\ p_2/p,\ldots,\ p_n/p)\). Since \(L^Q\) is a \(KB\)-space, it follows from this
* Throughout the note, the term reflexivity is understood in the sense of the theory of normed spaces.
(under the indicated conditions \(1 < p_i < \infty\)) the reflexivity of the spaces \(L^p\) follows.
Theorem 2. Let \(X\) be a separable continuous \(KB\)-space with unit, and let \(M(u)\) be an \(N\)-function which, together with the complementary \(N\)-function, satisfies the \(\Delta_2\)-condition. Put
\[ X_M=\{x:x\in X,\ M(x)\in X\} \]
and introduce on \(X_M\) a new norm by setting
\[ \|x\|_{(M)}=\inf\left\{K:K>0,\quad \left\|M\left(\frac{x}{K}\right)\right\|\leq 1\right\} \]
(here \(\|\ \|\) is the norm in \(X\)). Then \(X_M\), with the indicated norm, is a reflexive separable \(KB\)-space.
Let us note that if, for example, \(X=L[0,1]\), then \(X_M\) is the Orlicz space \(L_M^{*}\), equipped with the Luxemburg norm, equivalent to the Orlicz norm \((^1)\).
The proof of Theorem 2 is based on known theorems on the concrete representation of separable \(KB\)-spaces \((^3)\).
Theorem 3. Let \(X\) be a \(KB\)-space with unit. Then for every \(x\in X\) there exists a fundament* \(Y\) in \(X\) such that \(x\in Y\) and such that on \(Y\) one can introduce a new norm under which \(Y\) will be a reflexive \(KB\)-space.
This theorem follows from Theorem 1. Namely, without loss of generality one may assume that \(x\) is a bounded element, since otherwise this could be achieved by choosing a new unit. Then for \(Y\) one may take any of the spaces \(X_p\) with \(p>1\). \(X_p\subset X\), since \(X\) is a space with unit.
The following is somewhat more difficult to prove.
Theorem 4. Let \(X\) be an infinite-dimensional \(KB\)-space with unit. Then for every \(x\in X\) there exists a fundament \(Z\) in \(X\) such that \(x\in Z\) and such that \(Z\), with some new norm, is a non-reflexive \(KB\)-space.
From some results of \((^3)\) it is not hard to derive the following.
Let \(P\) be a \(KB\)-space with unit, on which an essentially positive linear functional \(f_0\) is given. Put
\[ Q=\left\{x:x\in \hat P,\ \sup_{\substack{0\leq x'\leq |x|\\ x'\in P}} f_0(x')<+\infty\right\}, \]
where \(\hat P\) is the maximal extension of the space \(P\). Denote by \(\bar f_0\) the completely linear extension of the functional \(f_0\) to all of \(Q\). It exists and is unique \((^3)\). Put also
\[ R=\{y:y\in Q,\ xy\in Q\ \text{for every } x\in P\}. \]
Then every functional \(f\in P'^{**}\) has the form
\[ f(x)=\bar f_0(xy), \tag{*} \]
where \(y\) is some element of \(R\), and such a representation is unique. The converse is also true: for every \(y\in R\), the functional \(f\) defined by formula \((*)\) belongs to \(P'\). Thus, in its composition, \(R\) may be regarded as the space conjugate to \(P\).
Theorem 5. In the space \(L[0,1]\) there exists a fundament \(R\) which contains all \(L_p[0,1]\) for \(p>1\) and which, for a certain choice of norm, is a reflexive \(KB\)-space.
\[ \text{* A linear subspace }Y\subset X\text{ is called a fundament if it is contained in }X\text{ normally and completely }((^2),\text{ p. }112). \]
\[ \text{** If }X\text{ is a normed space, then by }X'\text{ we denote the space conjugate to }X. \]
Let us give a brief outline of the proof of this theorem. Let \(P\) be the set of all functions from \(L[0,1]\) for which
\[ \|x\|=\left[\sum_{p=1}^{\infty}\frac{1}{p^2}\|x^2\|_{L_p}\right]^{1/2}<+\infty . \]
With the aid of Theorem 1 one can verify that \(P\) is a reflexive \(KB\)-space. Now we use the preceding assertion on the general form of a linear functional, taking
\[ f_0(x)=\int_0^1 x(t)\,dt, \]
where \(x\in P\). In this case \(Q\) coincides with \(L\). Then the space \(R\) will consist of all such \(y\in L\) that
\[ \int_0^1 |x(t)y(t)|\,dt<+\infty \]
for every \(x\in P\). This \(R\) is the required one, if it is endowed with the topology of the space conjugate to \(P\).
Remark 1. The constructed space \(R\), although it is a reflexive \(KB\)-space, is not a space of type \(X_p\) for any choice of the \(KB\)-space \(X\) and number \(p>1\).
Remark 2. D. A. Vladimirov drew my attention to the fact that \(R\) is not an Orlicz space, but contains all reflexive Orlicz spaces. Moreover, if \(x\) is an arbitrary element of \(R\), then all functions equimeasurable with \(x\) are also contained in \(R\).
Theorem 6. Let \(X\) be a \(K\)-space, and let \(X_1\) and \(X_2\) be two of its fundamentals that are reflexive \(KB\)-spaces with norms \(\|\ \|_1\) and \(\|\ \|_2\), respectively. Put \(X_3=X_1\cap X_2\). Introduce on \(X_3\) the norm
\[ \|x\|_3=\max\{\|x\|_1,\|x\|_2\}. \]
Then \(X_3\) is a reflexive \(KB\)-space.
Let us note that from this theorem there follows an answer to one of Halperin’s questions ((4), pp. 247—249). Namely, using Halperin’s notation, let us indicate that the space \(L^\lambda\), constructed from the function
\[ \lambda(u)=\lambda^p_{(\omega_1,\omega_2)}(u)=\max\bigl(\lambda^p_{(\omega_1)}(u),\lambda^p_{(\omega_2)}(u)\bigr), \]
is reflexive, if the spaces \(L^p_{(\omega_1)}\) and \(L^p_{(\omega_2)}\) are reflexive.
Theorem 7. Under the conditions of Theorem 6, on the set \(X_1+X_2\), i.e. in the linear hull of \(X_1\) and \(X_2\), one can define a norm in such a way that it will be a reflexive \(KB\)-space.
It follows from Theorems 6 and 7 that all fundamentals of a given \(K\)-space \(X\) that are reflexive \(KB\)-spaces form a distributive structure if ordered by inclusion. Denote this structure by \(\mathfrak{S}\). If \(X\) contains a unit \(e\), put
\[ \mathfrak{T}=\{Y:\ e\in Y\in\mathfrak{S}\}. \]
In this case \(\mathfrak{T}\) is a substructure of \(\mathfrak{S}\).
Let, in particular, \(X\) be a \(KB\)-space with an additive norm and a unit. Take the functional
\[ f_0(x)=\|x_+\|-\|x_-\| \]
and use the remark made earlier. Then to every reflexive \(KB\)-space \(Y\) that is a fundamental in \(X\) and contains the unit (that is, \(Y\in\mathfrak{T}\)), its conjugate \(Y'\) is assigned uniquely, and \(Y'\in\mathfrak{T}\). One can prove that in the distributive structure \(\mathfrak{T}\) there are
the following relations between conjugate spaces hold; namely, for any \(Y_1, Y_2 \in \mathfrak{S}\),
\[ (Y_1 \vee Y_2)' = Y_1' \wedge Y_2', \qquad (Y_1 \wedge Y_2)' = Y_1' \vee Y_2'. \]
If \(Y_1 = Y_1'\), then \(Y_1\) is a Hilbert space \(X_2\).
I express my gratitude to my scientific adviser, Prof. B. Z. Vulikh, for his attention and valuable advice.
Received
15 IV 1964
References
- M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.
- B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, 1961.
- L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Partially Ordered Spaces, 1950.
- J. Halperin, Proc. Symposium on Linear Spaces, Jerusalem, 1961.
- A. Benedek, R. Panzone, Duke Math. J., 28, No. 3, 301 (1961).