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MATHEMATICS
S. G. KREIN and Yu. I. PETUNIN
A CRITERION FOR THE RELATEDNESS OF TWO BANACH SPACES
(Presented by Academician A. N. Kolmogorov on 21 IV 1961)
In the paper (¹) the notion of related Banach spaces was introduced. Two Banach spaces \(E_0\) and \(E_1\) are called related if there exists a family of Banach spaces \(\{E_\alpha\}\) \((0 \leq \alpha \leq 1)\) with norms \(\|x\|_\alpha\), possessing the following properties:
1) for \(0 \leq \alpha < \beta \leq 1\), the space \(E_\beta\) is densely embedded in the space \(E_\alpha\), and
\[ \|x\|_\alpha \leq \|x\|_\beta \quad (x \in E_\beta); \tag{1} \]
2) for \(0 \leq \alpha < \beta < \gamma \leq 1\) and \(x \in E_\gamma\),
\[ \|x\|_\beta \leq \|x\|_\alpha^{\frac{\gamma-\beta}{\gamma-\alpha}} \|x\|_\gamma^{\frac{\beta-\alpha}{\gamma-\alpha}}; \]
3) for \(x \in E_1\),
\[ \lim_{\alpha \to 1} \|x\|_\alpha = \|x\|_1. \]
In the present paper a criterion is given for two Banach spaces to be related.
We shall say that a Banach space \(E_1\) is normally embedded in a Banach space \(E_0\) if \(E_1\) can be identified with some everywhere dense linear manifold in \(E_0\), in such a way that \(\|x\|_{E_0} \leq \|x\|_{E_1}\) \((x \in E_1)\).
Theorem 1. Let the Banach space \(E_1\) be normally embedded in the Banach space \(E_0\). In order that the spaces \(E_0\) and \(E_1\) be related, it is necessary and sufficient that the unit ball \(S_1\) of the space \(E_1\) be a closed set in the topology induced in \(E_1\) by the topology of the space \(E_0\).
Proof. Suppose that the spaces \(E_0\) and \(E_1\) are related, and let us show that the ball \(S_1\) has the required property. If the contrary is assumed, then there will exist a sequence \(x_n \in S_1\) and an element \(x \in S_1\) \((\|x\|_1 = a > 1)\) such that \(\|x_n - x\|_0 \to 0\). From 2) it follows that, for \(0 < \alpha < 1\),
\[ \|x_n - x\|_\alpha \leq \|x - x_n\|_0^{1-\alpha}\|x - x_n\|_1^\alpha \leq (a+1)^\alpha \|x - x_n\|_0^{1-\alpha} \to 0. \]
Thus the sequence \(x_n\) converges to \(x\) in all the spaces \(E_\alpha\) \((\alpha < 1)\). Using 3), one can choose \(\alpha_0\) so close to unity that \(\|x\|_{\alpha_0} = a - \varepsilon > 1\). But \(\|x_n\|_{\alpha_0} \leq \|x_n\|_1 \leq 1\), and, consequently, the sequence \(x_n\) cannot converge to \(x\) in the space \(E_{\alpha_0}\). We have arrived at a contradiction. The necessity is proved.
From the fact that the space \(E_1\) is densely embedded in the space \(E_0\), it follows that every linear functional \(\langle x, x'\rangle\), bounded in the norm of \(E_0\), is bounded also in the norm of \(E_1\). Thus the space \(E_0'\) conjugate to \(E_0\) is naturally regarded as a part of the space \(E_1'\). It is not difficult to show,
that \(E_0'\) is dense in \(E_1'\) in the weak topology \(\sigma(E_1', E_1)\). Moreover, from the inequality \(\|x\|_{E_0}\leq \|x\|_{E_1}\) \((x\in E_1)\) there follows the inequality \(\|x'\|_{E_1'}\leq \|x'\|_{E_0'}\).
Introduce on the linear manifold \(E_1\) a family of norms by the formula
\[ \|x\|_\alpha=\sup_{x'\in E_0'}\frac{|\langle x,x'\rangle|}{\|x'\|_0^{1-\alpha}\|x'\|_1^\alpha}. \tag{2} \]
The quantity \(\|x\|_\alpha\), obviously, has the properties of a norm. Further, for each \(x\in E_1\) and \(x'\in E_0'\), the function
\[ \frac{|\langle x,x'\rangle|}{\|x'\|_0^{1-\alpha}\|x'\|_1^\alpha} = \frac{|\langle x,x'\rangle|}{\|x'\|_0} \left(\frac{\|x'\|_0}{\|x'\|_1}\right)^\alpha \tag{3} \]
is continuous and logarithmically convex in \(\alpha\) on \([0,1]\). Since \(\|x'\|_1\leq \|x'\|_0\) \((x'\in E_0')\), the function (3) increases monotonically in \(\alpha\). It is then easy to see that the supremum of all the functions (3) over \(x'\in E_0'\), i.e. \(\|x\|_\alpha\), is a continuous logarithmically convex function of \(\alpha\). Thus the set \(E_1\), with the norms (2), forms a continuous incomplete normal scale of spaces with base \(E_1\) (see (1)). The family of spaces \(E_\alpha\) obtained by completing \(E_1\) in the norms (2) will have properties 1), 2).
For \(\alpha=0\)
\[ \|x\|_0=\sup_{x'\in E_0'}\frac{|\langle x,x'\rangle|}{\|x'\|_0^{1-\alpha}\|x'\|_1^\alpha} =\|x\|_{E_0}, \]
and, by virtue of the density of \(E_1\) in \(E_0\), the completion of \(E_1\) in the norm \(\|x\|_0\) coincides with the space \(E_0\).
If we show that
\[ \|x\|_1=\sup_{x'\in E_0'}\frac{|\langle x,x'\rangle|}{\|x'\|_0^{1-\alpha}\|x'\|_1^\alpha} =\|x\|_{E_1}, \tag{4} \]
then property 3) will hold for the family of spaces \(E_\alpha\), and the theorem will be proved.
Equality (4) is equivalent to the following:
\[ \sup_{x'\in E_0'\cap S_1'}\langle x,x'\rangle = \sup_{x'\in S_1'}\langle x,x'\rangle, \]
where \(S_1'\) is the unit ball in the space \(E_1'\). For the latter to hold it is necessary and sufficient that the regularly convex hull of the set \(E_0'\cap S_1'\), or, equivalently, its weak closure, coincide with \(S_1'\) (see \((^2)\)). In Bourbaki’s terminology (see \((^3)\), p. 275) this means that the characteristic of the set \(E_0'\) in the space \(E_1'\) must be equal to 1. For this it is necessary and sufficient (see \((^3)\), p. 275) that the ball \(S_1\) be closed in the topology \(\sigma(E_1,E_0')\). But, by virtue of the convexity of the set \(S_1\), its closure in the topology \(\sigma(E_1,E_0')\) will coincide with its closure in the topology induced by the topology of the space \(E_0\) on \(E_1\), and in this topology it is closed by assumption. The theorem is proved.
Corollary 1. If the space \(E_1\) is normally embedded in the space \(E_0\), and the conjugate space \(E_0'\) is dense in the conjugate space \(E_1'\), then the spaces \(E_0\) and \(E_1\), as well as the spaces \(E_0'\) and \(E_1'\), are related.
Corollary 2. A reflexive space \(E_1\) is related to any Banach space in which it can be normally embedded.
Corollary 2 follows from Corollary 1. However, the fulfillment of the conditions of Theorem 1 in this case follows from one result of the paper \((^4)\).
Corollary 3. Let \(E_0 \supset E_1 \supset E_2\) be three Banach spaces normally embedded in one another. If \(E_0\) and \(E_2\) are related, then \(E_1\) and \(E_2\) are also related.
Corollary 4. Let \(E_0 \supset E_1 \supset E_2\) be three Banach spaces normally embedded in one another. If \(E_1\) and \(E_2\) are related and the unit sphere of \(E_2\) is relatively compact in \(E_1\), then \(E_1\) and \(E_2\) are related.
Corollary 5. Let \(E_0\) be a reflexive Banach space, and let \(A\) be a linear operator with domain \(D_A\) dense in \(E_0\), acting in \(E_0\) and weakly closed. Introduce on \(D_A\) the norm \(\|x\|_1=\|x\|_0+\|Ax\|_0\). In this norm \(D_A\) will be a Banach space \(E_1\). The spaces \(E_0\) and \(E_1\) are related.
Let us consider some examples.
Let \(C(0,1)\) be the space of all functions \(x(t)\) continuous on \([0,1]\), with norm \(\|x\|_0=\max_{0\le t\le 1}|x(t)|\), and let \(C_1(0,1)\) be the space of all functions continuously differentiable on \([0,1]\), with norm \(\|x\|_1=\max_{0\le t\le 1}|x(t)|+\max_{0\le t\le 1}|x'(t)|\). It is not difficult to verify that for these spaces the conditions of Theorem 1 are satisfied and, consequently, they are related. Moreover, the unit sphere in \(C_1(0,1)\) is relatively compact in \(C(0,1)\). Therefore, by Corollary 4, the space \(C_1(0,1)\) will be related to any space into which the space \(C(0,1)\) can be normally embedded. For example, \(C_1(0,1)\) is related to each of the spaces \(L_p(0,1)\) \((1\le p<\infty)\).
Similarly one can show that the space \(L_p(0,1)\) and the space \(C_\alpha(0,1)\), consisting of functions satisfying on \([0,1]\) the Hölder condition with exponent \(\alpha\), are related.
Using Corollary 5, one can show that the Sobolev space \(W_p^{(l)}(D)\) with norm
\[ \|x\|_1= \left[\int_D |x(t)|^p\,dt\right]^{1/p} + \left\{ \int_D \left[ \sum_{k_1,k_2,\ldots,k_l=1}^{\mu} \left( \frac{\partial^l x}{\partial t_{k_1}\cdots \partial t_{k_l}} \right)^2 \right]^{p/2} dt \right\}^{1/p} \]
is related to the space \(L_p(D)\). Applying the theorem on the complete continuity of the embedding operator \(W_p^{(l)}(D)\) into \(L_p(D)\), on the basis of Corollary 4 one may assert that the space \(W_p^{(l)}(D)\) is related to all spaces \(L_q(D)\) \((1<q\le p)\) under the corresponding normalization. On the other hand, by Corollary 3, the space \(W_p^{(l)}(D)\) will be related to all spaces \(L_q(D)\) and \(W_q^{(m)}(D)\) for which the embedding theorems of S. L. Sobolev are valid.
If the spaces \(E_0\) and \(E_1\) are not related, then the question arises whether it is possible to introduce in \(E_1\) an equivalent norm under which it becomes related to \(E_0\).
Theorem 2. Let the Banach space \(E_1\) be normally embedded in the Banach space \(E_0\). In order that an equivalent norm can be introduced in the space \(E_1\), in which it will be related to the space \(E_0\), it is necessary and sufficient that the intersection of the closure in the space \(E_0\) of the unit ball \(S_1\) of the space \(E_1\) with the space \(E_1\) be a bounded set in it.
We give an example of two related spaces which, under an equivalent renorming, cease to be related. Take as \(E_0\) the space \(C(0,1)\), and as \(E_1\) the space \(C_1(0,1)\). These spaces, under the usual norm, are related. Introduce in \(E_1\) the norm by the formula
\[ \|x\|_1=\max_{0\le t\le 1}|x(t)|\max_{0\le t\le 1}|x'(t)|+|x'(1)|. \]
It is easy to see that the unit ball \(\|x\|_1\le 1\) is not closed in the sense of uniform convergence on \([0,1]\), and therefore the spaces \(E_0\) and \(E_1\) are not related.*
* The authors are indebted to M. A. Krasnosel’skii for this example.
For the construction of examples analogous to the one given, one may give the following general scheme. Let \(E_0\) and \(E_1\) be two normally embedded spaces, and let \(\langle x, x' \rangle\) be a functional from the space \(E'_1\) not belonging to the space \(E'_0\). Consider the hyperplane \(\langle x, x' \rangle=a\) \((0<a<1)\), which is an everywhere dense linear manifold in the space \(E_0\), and the balanced convex body \(W=\{x;\ \|x\|_1\leq 1,\ |\langle x,x'\rangle|\leq a\}\) of the space \(E_1\). It may happen that \(W\) is not closed in the topology of the space \(E_0\). Then the gauge function
\[ \rho(x)=\max\left\{\|x\|_1,\ \frac{1}{a}|\langle x,x'\rangle|\right\}, \]
defined by the body \(W\), will define on \(E_1\) a norm equivalent to the former one, while the space \(E_1\), endowed with this norm, is not related to the space \(E_0\).
Let us give an example of two normally embedded spaces which do not become related under any equivalent renorming of them. Denote by \(E_1\) the Banach space consisting of double numerical sequences \(x=(\xi_{nm})\) converging to zero, with norm
\[ \|x\|_1=\sup_{1\leq n,m<\infty}|\xi_{nm}|. \]
Let \(\{a_{nm}\}\) be a double sequence of positive numbers for which
\[ \sum_{n,m=1}^{\infty} a_{nm}=1. \]
Define the norm of \(x\) by the formula
\[ \|x\|_0=\sum_{n,m=1}^{\infty} a_{nm}\eta_{nm}, \]
where
\[ \eta_{nm}=\frac{|\xi_{nm}-n\xi_{n,m+1}|}{n+1}. \]
The norm \(\|x\|_0\) has the property that the closure, in this norm, of the unit ball \(S_1\) of the space \(E_1\) is unbounded in the metric of the space \(E_1\). By Theorem 2, the space \(E_0\), obtained from \(E_1\) by completion in the norm \(\|x\|_0\), will not be related to \(E_1\) if the space \(E_1\) is endowed with any norm equivalent to the norm \(\|x\|_1\).
In [1] it was shown that, for any two related spaces \(E_0\) and \(E_1\), one can construct a maximal continuous normal scale of spaces \(\{G_\alpha\}\) \((0\leq \alpha\leq 1)\). The space \(G_0\) coincides with \(E_0\), and if the spaces \(E_0\) and \(E_1\) are related, then \(G_1\) coincides with \(E_1\). If, however, \(E_0\) and \(E_1\) are not related, then the space \(E_1\) is normally embedded in \(G_1\). It follows from the preceding that the space \(G_1\) can be constructed as follows: the unit sphere \(S_1\) of the space \(E_1\) is closed in the space \(E_0\). The intersection of the closure \(\overline S_1\) with the space \(E_1\) is a convex body in \(E_1\). The gauge function constructed from this body defines a new norm in \(E_1\). The completion of \(E_1\) in this norm gives the space \(G_1\).
The authors express their gratitude to M. A. Krasnosel’skii and A. Yu. Levin for valuable discussions.
Voronezh State University
Received
20 II 1961
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