Abstract
Full Text
V. I. Gurarii
ON INCLINATIONS OF SUBSPACES AND CONDITIONAL BASES IN BANACH SPACE
(Presented by Academician A. N. Kolmogorov, 23 II 1962)
Let (P) and (Q) be linear manifolds in a Banach space (E). Define the inclination of (P) to (Q) as the quantity
[
\widehat{(P;\,Q)}=\inf_{\substack{x\in P,\; y\in Q\ |x|=1}}|x+y|.
]
Denote by (L_{{f_1,\ldots,f_n}}) the linear span of the elements (f_1,\ldots,f_n;\ f_i\in E) ((i=1,2,\ldots,n)), and call the index of the collection ({f_i}_{i=1}^n) the quantity
[
\gamma_{{f_1,\ldots,f_n}}=\min_{1\le p<n}\left(L_{{f_1,\ldots,f_p}};\widehat{\ }\,L_{{f_{p+1},\ldots,f_n}}\right).
]
Analogously, the index of a sequence ({f_i}_{i=1}^{\infty},\ f_i\in E), is defined as
[
\gamma_{{f_1,\ldots,f_n,\ldots}}=\inf_n \gamma_{{f_1,\ldots,f_n}}.
]
As was shown by M. M. Grinblum ((^{1})), in order that a sequence ({f_i}{i=1}^{\infty}) complete in (E) be a basis* in (E), it is necessary and sufficient that the condition (\gamma>0) be satisfied.
From the results of K. I. Babenko ((^{2})) and V. G. Vinokurov ((^{3})) it follows that in Hilbert spaces and in some classes of Banach spaces there exist subspaces with conditional bases. Here a generalization of this assertion to arbitrary infinite-dimensional Banach spaces will be obtained. Theorems 1–4 will serve as the apparatus for obtaining the results.
Theorem 1. For given (\varepsilon>0) and a finite-dimensional subspace (P\subset E) of an infinite-dimensional Banach space (E), there exists an infinite-dimensional subspace (Q\subset E) such that (\widehat{(P;\,Q)}>1-\varepsilon).
Proof. On the basis of the Banach–Mazur theorem one may regard (E) as a subspace of (C_{[0;1]}). The unit sphere (S_P) of the finite-dimensional subspace (P) is compact in (C_{[0;1]}) and, consequently, constitutes a family of functions equicontinuous on ([0;1]). Therefore, for the given (\varepsilon>0) there is a natural number (n) such that if (|x_1-x_2|<1/n), then (|f(x_1)-f(x_2)|<\varepsilon) for every (f(x)\in S_P).
Consider the set (Q) of functions from (E) that vanish at the points (x_k=k/n,\ k=0,1,\ldots,n). It is not difficult to verify that (Q) is an infinite-dimensional subspace of the space (E).
* A sequence ({f_i}_1^\infty) is called a basis in (E) if every element (f\in E) can be represented in a unique way in the form
[
f=\sum_{i=1}^{\infty}\alpha_i f_i .
]
A sequence ({f_i}_1^\infty) that remains a basis in (E) under any permutation of its elements is called an unconditional basis in (E). A basis that is not unconditional is called a conditional basis.
For the given (f(x) \in S_P) and (g(x) \in Q), let (\tilde x) be one of the points at which (|f(x)|) assumes its maximum; then for some natural (i_0 \leqslant n),
[
\frac{i_0-1}{n}\leqslant \tilde x \leqslant \frac{i_0}{n}.
]
We have:
[
\max_{x\in[0;1]} |f(x)-g(x)|
\geqslant
\max_{i=0,1,\ldots,n}\left|f\left(\frac{i}{n}\right)-g\left(\frac{i}{n}\right)\right|
=
]
[
=
\max_{i=0,1,\ldots,n}\left|f\left(\frac{i}{n}\right)\right|
\geqslant
\left|f\left(\frac{i_0}{n}\right)\right|
\geqslant |f(\tilde x)|-\varepsilon
=
1-\varepsilon .
]
This inequality means that ((\widehat{P;Q}) \geqslant 1-\varepsilon). The theorem is proved.
Lemma. Let, for linear manifolds (P_1, P_2=Q+R), and (P_3) in a Banach space, the following conditions be satisfied:
-
((\widehat{Q;R}) \geqslant \alpha > 0).
-
((P_1+\widehat{P_2;P_3}) \geqslant \beta;\quad (\widehat{P_1;P_2+P_3}) \geqslant \beta;\quad \beta>0.)
Then
[
(P_1+Q; R+P_3) \geqslant \frac{\alpha\beta}{2+\alpha}.
]
From the lemma there follows directly the following theorem on the possibility of “synthesizing” a basis in a space from bases in its subspaces.
Theorem 2. Let ({f_i^{(1)}, f_i^{(2)}, \ldots, f_i^{(k_i)}}), (i=1,2,\ldots), be bases in (k_i)-dimensional subspaces (P_i\subset E), and let
[
\gamma_{{f_i^{(1)},\ldots,f_i^{(k_i)}}} \geqslant \alpha >0,\quad i=1,2,\ldots,
]
and, for any natural (m,n), (m0.
]
Then the sequence
[
f_1^{(1)},\ldots,f_1^{(k_1)}, f_2^{(1)},\ldots,f_2^{(k_2)},\ldots, f_i^{(1)},\ldots,f_i^{(k_i)},\ldots
]
is a basis in its closed linear span, and
[
\gamma_{{f_1^{(1)},\ldots,f_1^{(k_1)},\ldots,f_2^{(1)},\ldots}}
\geqslant
\frac{\alpha\beta}{2+\alpha}.
]
The result obtained below makes essential use of an important theorem on (\varepsilon)-isometry of Banach spaces due to A. Dvoretzky ((^4)).
Definition. Banach spaces (E_1) and (E_2) are called (\varepsilon)-isometric if there exists an isomorphic mapping (T) from (E_1) onto (E_2) such that, for any (f\in E_1),
[
(1-\varepsilon)|f|\leqslant |Tf|\leqslant (1+\varepsilon)|f|.
]
Theorem 3 (A. Dvoretzky). For every (\varepsilon>0) and natural (n) there exists a natural (N_0(n;\varepsilon)) such that every (N)-dimensional Banach space, for (N>N_0), contains an (n)-dimensional subspace (\varepsilon)-isometric to (n)-dimensional Euclidean space.
From the cited results of K. I. Babenko, M. M. Grinblyum, and A. Dvoretzky it follows:
Theorem 4. In an infinite-dimensional Banach space (E), for every (\varepsilon>0) there can be found a collection of elements (f_1,\ldots,f_n), (n=n(\varepsilon)), such that
(\gamma_{{f_1,\ldots,f_n}}>\alpha>0), where (\alpha) is an absolute constant, and under a suitable permutation of the elements the index of this collection becomes less than (\varepsilon).
Theorem 5. In every infinite-dimensional Banach space there exists a sequence ({f_i}_{i=1}^{\infty}) which is a conditional basis in the closure of its linear span.
Proof. Consider a sequence of numbers ({\varepsilon_i}_{i=1}^{\infty}) satisfying the conditions: (0<\varepsilon_i<1) and
[
\prod_{i=1}^{\infty}(1-\varepsilon_i)=\beta>0.
]
By Theorem 4, in the given Banach space (E) there exists a collection of elements ({f_1^{(1)}, f_1^{(2)}, \ldots, f_1^{(k_1)}}) such that
[
\gamma_{{f_1^{(1)},\ldots,f_1^{(k_1)}}}>\alpha>0,
]
but, under a suitable permutation of its elements, we shall have
[
\gamma_{{\widetilde f_1^{(1)},\ldots,\widetilde f_1^{(k_1)}}}<\varepsilon_1.
]
Denote by (P_1) the subspace spanned by (f_1^{(1)}, \ldots, f_1^{(k_1)}), and consider an infinite-dimensional subspace (Q_1\subset E) such that
[
(P_1,Q_1)>1-\varepsilon_1
]
(the existence of such a subspace is ensured by Theorem 1).
In the subspace (Q_1) there exists a collection of elements ({f_2^{(1)}, \ldots, f_2^{(k_2)}}) such that
[
\gamma_{{f_2^{(1)},\ldots,f_2^{(k_2)}}}>\alpha,
]
but, under a suitable permutation of the elements, we shall have
[
\gamma_{{\widetilde f_2^{(1)},\ldots,\widetilde f_2^{(k_2)}}}<\varepsilon_2.
]
Denote by (P_2) the subspace spanned by (f_2^{(1)}, \ldots, f_2^{(k_2)}), and consider an infinite-dimensional subspace (Q_2\subset E) such that
[
(P_1+P_2,Q_2)>1-\varepsilon_2.
]
Continuing this process indefinitely, we obtain a collection of a sequence of finite-dimensional subspaces ({P_i}_{i=1}^{\infty}) with bases ({f_i^{(1)},\ldots,f_i^{(k_i)}}), for which we have
[
\gamma_{{f_i^{(1)},\ldots,f_i^{(k_i)}}}>\alpha
]
and
[
\gamma_{{\widetilde f_i^{(1)},\ldots,\widetilde f_i^{(k_i)}}}<\varepsilon_i
]
(the collection ({\widetilde f_i^{(1)}, \ldots, \widetilde f_i^{(k_i)}}) is obtained from the collection ({f_i^{(1)}, \ldots, f_i^{(k_i)}}) by some permutation of the elements), and, moreover,
[
(P_1+P_2+\cdots+P_i,Q_i)>1-\varepsilon_i,\qquad i=1,2,\ldots.
]
Since (P_{i+1}\subset Q_i), it follows that
[
(P_1+P_2+\cdots+P_i,P_{i+1})\ge (P_1+P_2+\cdots+P_i,Q_i)>1-\varepsilon_i.
]
Let us show that the sequence
[
{f_1^{(1)},\ldots,f_1^{(k_1)}, f_2^{(1)},\ldots,f_2^{(k_2)}, \ldots}
]
forms a basis in the subspace (P\subset E) spanned by it.
Let
[
f\in P_1+\cdots+P_m,\qquad
g\in P_{m+1}+\cdots+P_{m+n},\qquad
g=g_1+g_2+\cdots+g_n,
]
where (g_k\in P_{m+k}), (k=1,2,\ldots,n). We have:
[
\begin{aligned}
|f+g|
&=|(f+g_1+\cdots+g_{n-1})+g_n| \
&\ge (1-\varepsilon_{m+n-1})|f+g_1+\cdots+g_{n-1}| \
&\ge (1-\varepsilon_{m+n-1})(1-\varepsilon_{m+n-2})
|f+g_1+\cdots+g_{n-2}| \ge \cdots \
&\ge (1-\varepsilon_{m+n-1})(1-\varepsilon_{m+n-2})\cdots(1-\varepsilon_m)|f|
\ge \beta|f|.
\end{aligned}
]
This inequality means that
[
(P_1+\cdots+P_m,\,P_{m+1}+\cdots+P_{m+n})\ge \beta,
]
and, by Theorem 2, the sequence
[
f_1^{(1)},\ldots,f_1^{(k_1)}, f_2^{(1)},\ldots,f_2^{(k_2)}, \ldots
]
is a basis in (P). This basis is conditional, since by a permutation of its elements one can obtain the sequence
[
\widetilde f_1^{(1)},\ldots,\widetilde f_1^{(k_1)}, \widetilde f_2^{(1)},\ldots,\widetilde f_2^{(k_2)}, \ldots,
]
for which, in view of the fact that (\lim_{i\to\infty}\varepsilon_i=0), we have:
[
\gamma_{{\widetilde f_1^{(1)},\ldots,\widetilde f_1^{(k_1)}, \widetilde f_2^{(1)},\ldots,\widetilde f_2^{(k_2)},\ldots}}=0.
]
The theorem is proved.
Kharkov Automobile and Highway Institute
Received
23 II 1962
References
- M. M. Grinblyum, DAN, 31, 428 (1941).
- K. I. Babenko, DAN, 62, 157 (1948).
- V. G. Vinokurov, DAN, 85, 685 (1952).
- A. Dvoretzky, Proc. Nat. Acad. Sci. USA, 45, 223 (1959).