Abstract Generated abstract
This paper studies finite-dimensional subspaces of Banach spaces with the e-property, meaning that any norm-one linear extension of the identity on the subspace must be the identity on the whole space. It gives criteria for saturated subspaces, which imply the e-property, in the spaces l_p^n, distinguishing the cases where p is or is not an even integer, and derives existence and nonexistence results for two-dimensional and higher-dimensional saturated subspaces. The paper also treats l_p and L_p spaces, proving the existence of certain two-dimensional saturated subspaces for non-even or odd integer p and showing nonexistence in some even-integer cases. Finally, it introduces quasi-smoothness as a broader sufficient condition for the e-property and relates the existence of finite-dimensional saturated subspaces to the Lazar and Zippin class A_1.
Full Text
UDC 513.88:513.83:513.82
MATHEMATICS
V. S. RUBLEV
ON A SPECIAL CLASS OF FINITE-DIMENSIONAL SUBSPACES OF A BANACH SPACE
(Presented by Academician A. N. Kolmogorov on 11 VI 1968)
Let \(E\) be a real Banach space and \(E_0\) some subspace of it. Suppose that from the conditions
\[ \|Ax\|\leq \|x\|\quad (x\in E),\qquad Ax \equiv x\quad (x\in E_0), \tag{1} \]
where \(A\) is a linear operator, there follows the identity
\[ Ax \equiv x\quad (x\in E). \tag{2} \]
Then we shall say that the subspace \(E_0\) has the \(e\)-property. A number of theorems on subspaces possessing the \(e\)-property were established in papers \((^{1,2})\). Of main interest to us is the fact that in many infinite-dimensional spaces there are finite-dimensional subspaces possessing the \(e\)-property. Such subspaces arise naturally in a number of problems in approximation theory.
Of interest is the problem of as complete as possible a description of subspaces possessing the \(e\)-property in various concrete Banach spaces, and the problem of computing or estimating the minimal dimension of such subspaces. In the present paper these problems are solved for the spaces \(l_p^n\), \(l_p\), and \(L_p\).
- Let \(S\) and \(S^*\) be the unit spheres respectively in the Banach space \(E\) and in its conjugate \(E^*\). One says 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 point of smoothness of \(S\) if through it there passes a unique functional \(f_0\in S^*\). One says that a subspace \(E_0\) of the space \(H\) is saturated with points of smoothness of \(S\) if the set \(F(E_0)\) of functionals passing through points of smoothness of \(S\) lying in \(E_0\) is total. Such subspaces are called saturated (see \((^{1,2})\)).
A saturated subspace has the \(e\)-property (cf. \((^{1,2})\)). Indeed, let \(A\) be an extension of the identity operator from \(E_0\) to \(E\) without increasing the norm (\(\|A\|=1\)). For any functional \(f_0\in F(E_0)\) there is a point \(x_0\in E_0\) such that \(f_0(x_0)=1\). Consider the functional \(g_0(x)=f_0(Ax)\) \((x\in E)\). Since \(|g_0(x)|=|f_0(Ax)|\leq \|Ax\|\leq \|x\|\) and \(g_0(x_0)=f_0(Ax_0)=f_0(x_0)=1\), it follows that \(g_0\in S^*\) and passes through the point of smoothness \(x_0\). Consequently, \(g_0\equiv f_0\), and \(f_0(Ax-x)=0\) for \(x\in E\). The totality of the set \(F(E_0)\) entails \(Ax\equiv x\) \((x\in E)\).
- A description of saturated subspaces in finite-dimensional spaces with metric \(l_p\) \((1\leq p<\infty)\) is given by the following theorems.
Theorem 1. Let \(p\) not be an even integer. Then, in order that in the space \(l_p^n\) the linearly independent vectors
\[ e_1=\{\xi_{11},\xi_{12},\ldots,\xi_{1n}\};\ldots; e_k=\{\xi_{k1},\xi_{k2},\ldots,\xi_{kn}\} \tag{3} \]
form a basis of a saturated subspace, it is necessary and sufficient that the \(k\)-dimensional vectors
\[ g_1=\{\xi_{11},\xi_{21},\ldots,\xi_{k1}\};\ldots; g_n=\{\xi_{1n},\xi_{2n},\ldots,\xi_{kn}\} \tag{4} \]
be pairwise non-collinear.
Thus, for example, the subspace spanned by the vectors
\[ e_1=\{0,1,\ldots,n(n-1)/2\};\qquad e_2=\{0,1,\ldots,(n-1)^2\}; \]
\[ e_3=\{1,3,\ldots,3^{\,n-1}\} \]
of the space \(l_p^n\) \((n\geq 4,\ 1\leq p<\infty,\ p\) is not an even integer\()\) is ...
saturated subspace, while the subspace spanned by any two of the indicated vectors is not a saturated subspace.
It follows from Theorem 1 that the vectors
\[ e_1=\{1,2,\ldots,n\},\qquad e_2=\{1,2^2,\ldots,n^2\} \tag{5} \]
form a basis of a two-dimensional saturated subspace in \(l_p^n\), if \(p\) is not an even integer.
Let now \(p\) be an even integer. In this case the space \(l_p^n\) may have no proper saturated subspaces. For example, the space \(l_2^n\) is Hilbert, and therefore it has no saturated subspaces. Below, by \(\varphi_j\) \((j=1,\ldots,C_{p+k-2}^{k-1})\) we denote all distinct homogeneous polynomials
\[ \varphi_j(x_1,\ldots,x_k)=x_1^{l_1}\cdots x_k^{l_k} \left(0\leq l_m\leq p-1,\ \sum_{m=1}^{k}l_m=p-1\right) \tag{6} \]
of degree \(p-1\) in \(k\) variables \(x_1,\ldots,x_k\).
Theorem 2. Let \(p\) be an even integer. In order that \(k\)-dimensional saturated subspaces exist in the space \(l_p^n\), it is necessary and sufficient that the condition
\[ C_{p+k-2}^{k-1}\geq n. \tag{7} \]
be satisfied.
From this theorem, in particular, it follows that in \(l_p^n\) (\(p\) an even integer) two-dimensional saturated subspaces exist if and only if \(p\geq n\).
Theorem 3. Let \(p\) be an even integer. Then, in order that in the space \(l_p^n\) linearly independent vectors
\[ e_1=\{\xi_{11},\xi_{12},\ldots,\xi_{1n}\},\ldots, e_k=\{\xi_{k1},\xi_{k2},\ldots,\xi_{kn}\} \tag{8} \]
form a basis of a saturated subspace, it is necessary and sufficient that the rank of the matrix with elements
\[ a_{ij}=\varphi_j(\xi_{1i},\xi_{2i},\ldots,\xi_{ki}) \qquad (i=1,\ldots,n;\ j=1,\ldots,C_{p+k-2}^{k-1}) \tag{9} \]
be equal to \(n\).
For example, if \(n\leq 4\), then the vectors (5) form a basis of a saturated subspace in \(l_4^n\).
We give one more assertion.
Theorem 4. If in \(l_p^n\) \((1\leq p<\infty)\) there exist two-dimensional saturated subspaces, then every saturated subspace contains a two-dimensional saturated subspace.
- We pass to the consideration of the space \(l_p\).
Theorem 5. Let \(p\) not be an even integer, \(1\leq p<\infty\). Let
\[ e_1=\{\xi_1,\xi_2,\ldots\},\qquad e_2=\{\eta_1,\eta_2,\ldots\}. \tag{10} \]
Suppose that the sequence of two-dimensional vectors
\[ \{\xi_i,\eta_i\}\qquad (i=1,2,\ldots) \tag{11} \]
contains no pairwise collinear vectors and that every subsequence of the sequence of numbers
\[ \alpha_i=\xi_i/\eta_i\qquad (i=1,2,\ldots) \tag{12} \]
of the extended number line contains isolated points (of this same subsequence).
Then the vectors (10) form a basis of a two-dimensional saturated subspace in the space \(l_p\).
Thus, for example, the subspace spanned by the vectors
\[ e_1=\left\{\frac12,\ldots,\frac{1}{2^n},\ldots\right\},\qquad e_1=\left\{\frac13,\ldots,\frac{1}{3^n},\ldots\right\} \]
is a saturated subspace of \(l_p\) \((1\leq p<\infty)\), if \(p\) is not an even integer. Thus, in the space \(l_p\) for \(p\), not
which is an even integer, there always exists a two-dimensional saturated subspace.
In the case where \(p\) is an even integer, it follows from the results obtained in (3) that in \(l_p\) there are no finite-dimensional saturated subspaces. However, the following is true.
Theorem 6. Let \(p \ne 2\). Then in \(l_p\) there are saturated subspaces of any finite defect.
- The following theorem establishes the existence of two-dimensional saturated subspaces in some spaces \(L_p\) \((1 \le p < \infty)\).
Theorem 7. Let \(p\) be an odd integer. Let \(u(t), v(t) \in L_p[0,1]\). Let the function \(\varphi(t)=u(t)/v(t)\) be defined and finite almost for all \(t \in [0,1]\), and suppose, moreover, that the following conditions are satisfied:
1) for some set \(D_0 \subseteq [0,1]\) of full measure the function \(\varphi(t)\) establishes a one-to-one correspondence between the points of \(D_0\) and \(D^0=\varphi(D_0)\).
2) there exists a finite or countable system of intervals \(\{a_k,b_k\}\) such that \(D_0\) is contained in their union and on each of the sets \(\{a_k,b_k\}\cap D_0\) the function \(\varphi(t)\) is strictly monotone;
3) there exists a finite or countable system of intervals \(\{c_j,d_j\}\) such that \(D^0\) is contained in their union and on each of the sets \(\{c_j,d_j\}\cap D^0\) the function \(\psi(s)\), inverse to the function \(\varphi(t)\), is strictly monotone.
Then the functions \(u(t)\) and \(v(t)\) form a basis of a two-dimensional saturated subspace in the space \(L_p[0,1]\).
We note that the conditions of Theorem 7 are satisfied for continuous \(u(t)\) and \(v(t)\) if, in particular, the function \(\varphi(t)\) is strictly monotone. It follows from Theorem 7, for example, that the subspaces spanned by the vectors \(u(t)\equiv 1\), \(v(t)\equiv t\), or \(u(t)\equiv \sin \pi t\), \(v(t)\equiv \cos \pi t\), are saturated subspaces in \(L_p[0,1]\), where \(p\) is an odd integer.
In (3) it was shown that in the case of an even integer \(p\) there are no saturated finite-dimensional subspaces in the spaces \(L_p\).
- We describe a broader class of subspaces possessing the \(e\)-property. We shall call a point \(x \in S\) a point of quasi-smoothness if the functionals \(f_x \in S^*\) passing through it form a compact set \(F_x\). Let \(M_0\) be the set of points of quasi-smoothness of \(S\) lying in the subspace \(E_0\). Denote by \(\mathfrak F(E_0)\) the collection of sets of functionals from \(S^*\), each of which contains, for one functional each, the sets \(F_x\) \((x\in M_0)\).
Theorem 8. Let each set in \(\mathfrak F(E_0)\) be total. Then \(E_0\) has the \(e\)-property.
The question remains open of necessary and sufficient conditions (in terms of properties of the set \(S\)) for the subspace \(E_0\) to have the \(e\)-property.
- Following A. Lazar and M. Zippin (see (3)), denote by \(A_1\) the class of Banach spaces \(E\) possessing the following property: for every finite-dimensional subspace \(E_0 \subset E\) there exists a subspace \(F \subset E^*\) of infinite defect such that
\[ \|x\|=\sup_{f\in F\cap S^*}|f(x)| \qquad (x\in E_0). \tag{13} \]
Obviously, every space in \(A_1\) contains no finite-dimensional saturated subspaces.
Theorem 9. Let the Banach space \(E\) with smooth sphere \(S\) not belong to the class \(A_1\). Then \(E\) contains finite-dimensional saturated subspaces.
In conclusion the author takes the opportunity to express gratitude to M. A. Krasnosel’skii, A. Yu. Levin, P. P. Zabreiko, and B. S. Mityagin for discussion of the work and advice.
Voronezh State University
Received
22 VI 1968
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- A. Lazar, M. Zippin, Israel J. Math., 3, no. 3, 147 (1965).