Abstract
Full Text
UDC 513.88+513.83+513.82
V. S. RUBLEV
ON SATURATED AND COMPLETELY SATURATED SUBSPACES OF CERTAIN COORDINATE SPACES
(Presented by Academician L. V. Kantorovich on 6 I 1969)
In connection with certain problems of analysis and, first of all, approximation theory, interest arose in describing subspaces \(E_0\) of a Banach space \(E\) having the property that from the condition
\[ Ax \equiv x \quad (x \in E_0), \tag{1} \]
where \(A\) is a linear operator leaving invariant some convex set \(V\), there follows the identity
\[ Ax \equiv x \quad (x \in E). \tag{2} \]
Descriptions of certain classes of such subspaces in the case when \(V\) is a cone in the space \(E\) are contained in works \((^{1-5})\), etc. In the case when \(V\) is the unit ball of the space \(E\), such subspaces (we called them subspaces possessing the \(e\)-property) were studied in works \((^{4-6})\). Thus, in \((^6)\) a description was given of a certain class of finite-dimensional subspaces possessing the \(e\)-property in the spaces \(l_p^n\), \(l_p\), and \(L_p\).
Alongside this, it is of interest to single out the following class of subspaces possessing stronger properties. Let \(E_0\) be a subspace of a Banach space \(E\), and suppose that from the conditions
\[ \|A_n x\| \leq \|x\| \quad (x \in E), \qquad \lim_{n\to\infty}\|A_n x - x\| = 0 \quad (x \in E_0), \tag{3} \]
where \(A_n\) is a sequence of linear operators, there follows the identity
\[ \lim_{n\to\infty}\|A_n x - x\| = 0 \quad (x \in E). \tag{4} \]
Then we shall say that the subspace \(E_0\) possesses the complete \(e\)-property. A number of theorems on subspaces possessing the complete \(e\)-property were established in works \((^{4,5})\).
In the present article an attempt is made to describe finite-dimensional subspaces possessing the \(e\)-property and the complete \(e\)-property in the coordinate spaces \(c\), \(m^n\), \(\lambda_\alpha^n\), where \(\lambda_\alpha^n\) is the \(n\)-dimensional space with the Lorentz norm \((^7)\).
- Important classes of subspaces possessing the \(e\)-property and the complete \(e\)-property are, respectively, the classes of saturated and completely saturated subspaces (see \((^{4-6})\)). Let us recall their definition.
Let \(S\) and \(S^*\) be the unit spheres, respectively, in the Banach space \(E\) and its conjugate \(E^*\). It is said that a functional \(f \in S^*\) passes through a point \(x \in S\) if \(f(x)=1\). A point \(x_0 \in S\) is called a smooth point of \(S\) if through it there passes a unique functional \(f_0 \in S^*\). It is said that a subspace \(E_0\) of the space \(E\) is saturated by smooth points of \(S\) if the set \(F(E_0)\) of functionals passing through smooth points of \(S\) lying in \(E_0\) is total. Such subspaces are called saturated.
It is said that \(E_0\) is completely saturated if the following conditions are fulfilled: a) \(E_0\) is a saturated subspace; b) the set \(F(E_0)\) contains such a total subset \(F_0(E_0)\) that the norm defined by the equality
\[ \|x\|^*=\sup_{f\in F_0(E_0)} |f(x)|, \tag{5} \]
is equivalent to the original norm of the space \(E\); c) every sequence of functionals \(f_n\in F_0(E_0)\) has a limit point \(f_0\in F(E_0)\) in the weak topology.
- Let us first consider the \(n\)-dimensional space \(m^n\) with norm \(\|x\|=\max_{1\le i\le n}|\xi_i|\).
Theorem 1. For linearly independent vectors
\[ e_1=\{\xi_{11},\ldots,\xi_{1n}\},\ldots,e_k=\{\xi_{k1},\ldots,\xi_{kn}\} \tag{6} \]
to form a basis of a saturated subspace \(E_k\) in \(m^n\), it is necessary and sufficient that each point of the collection \(M_k=\{\pm g_i\}_1^n\), where
\[ g_i=\{\xi_{1i},\xi_{2i},\ldots,\xi_{ki}\}\quad (i=1,\ldots,n), \tag{7} \]
not belong to the convex hull of the others.
It follows from Theorem 1 that two-dimensional saturated subspaces exist in the space \(m^n\). Indeed, if \(M_2\) is a plane centrally symmetric convex \(2n\)-gon, then to every collection of \(n\) distinct vertices \(\{x_i,y_i\}\) \((i=1,\ldots,n)\) of it, containing no opposite vertices, there correspond vectors
\[ \{x_1,x_2,\ldots,x^n\}\in m^n,\qquad \{y_1,y_2,\ldots,y_n\}\in m^n, \]
forming a basis of a two-dimensional saturated subspace of the space \(m^n\).
Let us note that every \(k\)-dimensional subspace containing a saturated subspace of smaller dimension is saturated. If, however, a subspace does not contain saturated subspaces of minimal dimension, then the question of whether it can be saturated is resolved differently for various concrete spaces. Thus, in the space \(l_p^n\) (see \((^6)\)) there are no saturated subspaces that do not contain saturated subspaces of minimal dimension, while in the space \(m^n\) such subspaces may exist. For example, the subspace \(E_3\) spanned by the vectors
\[ e_1=\{1,1,1,1\},\qquad e_2=\{1,1,-1,-1\},\qquad e_3=\{1,-1,1,-1\}, \]
is a saturated subspace of the space \(m^4\), since the convex hull \(M_3\) is a cube in three-dimensional space and the conditions of Theorem 1 are fulfilled. However, none of the projections of the cube can be an octagon, and therefore \(E_3\), as follows from the following Theorem 2, does not contain two-dimensional saturated subspaces of the space \(m^4\).
Theorem 2. For a saturated subspace \(E_k\) of the space \(m^n\), spanned by the vectors (6), to contain a saturated subspace \(E_l\) of smaller dimension \((1<l<k)\), it is necessary and sufficient that there exist a projection of the convex hull \(M_k\) onto some \(l\)-dimensional subspace having the same number of vertices.
- Let us turn to the consideration of the space \(c\) of convergent sequences, in which the norm is defined by the equality \(\|x\|=\sup_i|\xi_i|\).
Introduce the notation
\[ \xi_0(x)=\xi_0=\lim_{i\to\infty}\xi_i\qquad (x=\{\xi_1,\xi_2,\ldots\}\in c). \tag{8} \]
Theorem 3. Let \(E_k\) be a \(k\)-dimensional subspace of the space \(c\), and let the vectors
\[ e_1=\{\xi_{11},\xi_{12},\ldots\},\ldots,e_k=\{\xi_{k1},\xi_{k2},\ldots\} \tag{9} \]
form a basis in \(E_k\).
\(E_k\) is a saturated subspace if and only if all vectors of the set \(M=\{\pm g_i\}_0^\infty\), where
\[ g_i=\{\xi_{1i},\xi_{2i},\ldots,\xi_{ki}\}\qquad (i=0,1,2,\ldots), \tag{10} \]
are distinct extreme points of the convex hull of \(M\).
Thus, for example, the subspace spanned by the vectors
\[ e_1=\left\{-1,0,\frac12,\ldots,\frac{n-2}{n-1},\ldots\right\}, \]
\[ e_2=\left\{1,2,1+\frac{\sqrt3}{2},\ldots,1+\frac{\sqrt{2n-3}}{n-1},\ldots\right\}, \tag{11} \]
is a saturated subspace in \(c\).
From the results obtained in (8), it follows that in the space \(c_0\) of sequences converging to zero there are no finite-dimensional saturated subspaces. We note that \(c_0\) likewise contains no finite-dimensional saturated subspaces of the space \(c\), although it is itself a saturated subspace in \(c\).
A description of completely saturated finite-dimensional subspaces of the space \(c\) is given by Theorem 4. By an exposed point \(g\) of a convex set \(G\) one means an extreme point of \(G\) through which there passes a hyperplane supporting \(G\) and having with \(G\) only \(g\) in common (see (9)).
Theorem 4. Let \(E_k\) be a \(k\)-dimensional subspace of the space \(u\), and let the vectors (9) form a basis in \(E_k\).
\(E_k\) is a completely saturated subspace if and only if all vectors of the set \(M=\{\pm g_i\}_0^\infty\), where \(g_i\) is defined by formula (10), are distinct exposed points of the convex hull of \(M\).
Thus, for example, the subspace spanned by the vectors
\[ e_1=\left\{0,\frac12,\ldots,\frac{n-1}{n},\ldots\right\},\qquad e_2=\left\{2,1+\frac{\sqrt3}{2},\ldots,1+\frac{\sqrt{2n-1}}{n},\ldots\right\} \]
obtained from the vectors (11) by a shift by one coordinate, is completely saturated, whereas the subspace spanned by the vectors (11) is not a completely saturated subspace of the space \(c\).
4. Consider the \(n\)-dimensional spaces with the Lorentz metric \(\lambda_\alpha^n\) \((0<\alpha<1)\). Let \(x=\{\xi_1,\xi_2,\ldots,\xi_n\}\in\lambda_\alpha^n\). The norm in the space \(\lambda_\alpha^n\) is given by the formula
\[ \|x\|_{\lambda_\alpha^n}=\int_0^\infty \varphi^\alpha(x,h)\,dh, \tag{12} \]
where \(\varphi(x,h)=\operatorname{mes}\{s:\ |\xi_s|\ge h\}\).
The following Theorem 5 shows that saturated subspaces form, in a certain sense, a dense set in the set of all subspaces of the space \(\lambda_\alpha^n\).
Theorem 5. Let \(E_0\) be a subspace of \(\lambda_\alpha^n\) of dimension greater than \(1\), and let \(\varepsilon\) be an arbitrary positive number.
There exists a saturated subspace \(E_0'\) in \(\lambda_\alpha^n\) such that the gap (see (10)) between the subspaces \(E_0\) and \(E_0'\) satisfies the inequality \(\theta(E_0,E_0')<\varepsilon\).
In the cases \(k=3\) and \(n=4\) one can give a complete description of saturated subspaces of the space \(\lambda_\alpha^n\). Thus, in \(\lambda_\alpha^4\) every subspace of dimension greater than \(1\) that contains at least one point of smoothness of the unit
of the sphere \(S\), is saturated. For \(n=3\) the exceptional case is that in which \(\alpha=\alpha_0=\log_3 2\). There, if \(\alpha\ne\alpha_0\), then again every two-dimensional subspace in \(\lambda_\alpha^3\) containing at least one smooth point of the unit sphere is saturated. If, however, \(\alpha=\alpha_0\), then in order that the two-dimensional subspace spanned by the vectors
\[ e_1=\{\xi_{11},\ \xi_{12},\ \xi_{13}\},\qquad e_2=\{\xi_{21},\ \xi_{22},\ \xi_{23}\} \]
be saturated, it is necessary and sufficient that for the vectors \(g_i=\{\xi_{1i},\ \xi_{2i}\}\) \((i=1,2,3)\) and arbitrary \(\delta_i\), taking one of the values \(0,1,-1\), from the condition \(\sum_{i=1}^{3}|\delta_i|>0\) it follow that \(\sum_{i=1}^{3}\delta_i g_i\ne0\).
Thus, for example, the subspace spanned by the vectors
\[ e_1=\{1,\ 0,\ -1\},\qquad e_2=\{0,\ 1,\ 0\} \]
is not saturated in \(\lambda_\alpha^3\) for any value \(\alpha\in(0,1)\); the subspace spanned by the vectors
\[ e_1'=\{1,\ 0,\ -1\},\qquad e_2'=\{1,\ 1,\ 0\}, \]
is saturated in \(\lambda_\alpha^3\) for all values \(\alpha\in(0,1)\), except \(\alpha=\alpha_0\), and the subspace spanned by the vectors
\[ e_1''=\{1,\ 0,\ -1\},\qquad e_2''=\{1,\ 1,\ 1\}, \]
is saturated in \(\lambda_\alpha^3\) for all values \(\alpha\in(0,1)\).
In conclusion the author takes the opportunity to express his gratitude to M. A. Krasnosel’skii, A. Yu. Levin, and P. P. Zabreiko for discussing the work and for their advice.
Voronezh State University
Received
30 XII 1968
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