UDC 513.88:513.83

N. I. GURARII

SOME THEOREMS ON BASES IN HILBERT AND BANACH SPACES

(Presented by Academician L. V. Kantorovich on 23 I 1970)

The purpose of the present note is to study sequences of coefficients in expansions of elements of uniformly convex or uniformly smooth (for the definition see, for example, \((^5)\)) Banach spaces \(E\) with respect to arbitrary seminormalized* bases \(\{e_k\}_1^\infty\) in \(E\). The case of a Hilbert space is considered separately.

§ 1. Theorem 1**. Let \(\{e_k\}_1^\infty\) be a seminormalized basis in a Banach space \(E\). Then:

a) If \(E\) is uniformly convex, then there exist numbers \(A>0\) and \(r>1\), depending on the basis \(\{e_k\}_1^\infty\), such that in the expansion of any element \(x \in E\),

\[ x=\sum_1^\infty \alpha_k e_k, \]

the inequality holds
\[ \|x\|\leq A\left(\sum_1^\infty |\alpha_k|^r\right)^{1/r}. \tag{1} \]

b) If \(E\) is uniformly smooth, then there exist numbers \(B>0\), \(s<\infty\), depending on the basis \(\{e^k\}_1^\infty\), such that in the expansion of any element \(x \in E\),

\[ x=\sum_1^\infty \alpha_k e_k, \]

the inequality holds
\[ \|x\|\geq B\left(\sum_1^\infty |\alpha_k|^s\right)^{1/s}. \tag{2} \]

Corollary. If a Banach space \(E\) is simultaneously uniformly convex and uniformly smooth, then there exist numbers \(r, s, A>0, B>0\), \(1<r\leq s<\infty\), depending on the basis \(\{e_k\}_1^\infty\), such that in the expansion of any element \(x \in E\),

\[ x=\sum_1^\infty \alpha_k e_k, \]

the inequality holds
\[ B\left(\sum |\alpha_k|^s\right)^{1/s}\leq \|x\|\leq A\left(\sum_1^\infty |\alpha_k|^r\right)^{1/r}. \tag{3} \]

Let us apply Theorem 1 to the study of the question of stability of bases in uniformly smooth Banach spaces. We first introduce the following

Definition 1. A complete sequence \(\{e_k\}_1^\infty\) in a Banach space \(E\) is called completely stable with respect to a positive sequence \(\{\varepsilon_k\}_1^\infty\) if for any po-

* A sequence \(\{e_k\}_1^\infty \subset E\) is called a basis in \(E\) if every element \(x\in E\) can be represented in a unique way in the form
\[ x=\sum_1^\infty \alpha_k e_k. \]
The basis \(\{e_k\}_1^\infty\) is called seminormalized if, for some \(m>0\), \(M>0\), the relation \(m\leq \|e_k\|\leq M,\ k=1,2,\ldots\), holds.

** This theorem was obtained jointly with V. I. Gurarii.

sequences \(\{g_k\}_1^\infty\) satisfying the condition \(\|g_k-e_k\|<\varepsilon_k,\ k=1,2,\ldots\), the linear operator \(T\), defined by the relation \(Te_k=g_k,\ k=1,2,\ldots\), can be represented in the form

\[ T=I+S,\qquad \|S\|<1. \]

Remark. Under the conditions of Definition 1, \(T\) is an isomorphism of \(E\) onto itself \((^6)\). Therefore, if \(\{e_k\}_1^\infty\) is a basis, then \(\{g_k\}_1^\infty\) is also a basis in \(E\), i.e., from complete stability there follows the usual stability of the basis, as well as the stability of other linear-topological properties of the sequence.

Theorem 2. Let \(\{e_k\}_1^\infty\) be a quasi-normalized basis in a uniformly smooth Banach space \(E\). Then there exist numbers \(p>1\) and \(R>0\), depending only on the basis \(\{e_k\}_1^\infty\), such that for any positive sequence \(\{\varepsilon_k\}_1^\infty\) satisfying the condition \(\sum_1^\infty \varepsilon_k^p<R\), \(\{e_k\}_1^\infty\) is completely stable relative to \(\{e_k\}_1^\infty\).

Proof. Choose the number \(p\) as conjugate, in the sense of Hölder, to \(s\), where \(s\) is defined in (2), i.e. \(p=s/(s-1)\). Let \(\{g_k\}_1^\infty\subset E\) be such that \(\|e_k-g_k\|\le \varepsilon_k,\ k=1,2,\ldots\). Define on the linear span \(L(\{e_k\}_1^\infty)\) of the sequence \(\{e_k\}_1^\infty\) a linear operator \(T\) by the equalities \(Te_k=g_k,\ k=1,2,\ldots\). Let \(T-I=S\); then \(S\) is also a linear operator, for the moment defined on \(L\). Estimate \(\|S\|\), putting \(x=\sum_1^n \alpha_k e_k\), and applying (2) and Hölder’s inequality:

\[ \|Sx\|=\|(T-I)x\|=\|Tx-x\|=\left\|\sum_1^n \alpha_k(g_k-e_k)\right\|\le \]

\[ \le \sum_1^n |\alpha_k|\varepsilon_k \le \left(\sum_1^n |\alpha_k|^s\right)^{1/s} \left(\sum_1^n \varepsilon_k^p\right)^{1/p} \le \frac{\|x\|}{B} R^{1/p}. \]

We shall assume \(R\) chosen so that \(R^{1/p}/B<1\). Then \(\|S\|<1\), and consequently the operator \(T\) is an isomorphism of \(L(\{e_k\}_1^\infty)\) onto \(L(\{g_k\}_1^\infty)\). Extending the bounded operator \(T\) to all of \(E\) by continuity, we obtain a linear operator \(\widetilde T\), which is an isomorphism of \(E\) onto itself and such that

\[ \widetilde T e_k=g_k,\qquad k=1,2,\ldots,\qquad \widetilde T=I+S,\qquad \|S\|<1. \]

The theorem is proved.

As far as we know, Theorem 2 is new also for the special case of Hilbert space, improving the condition \(\sum_1^\infty \varepsilon_k<R\) in the theorem of Krein–Milman–Rutman \((^4)\).

§ 2. The starting point of the results of this section is inequality (3). A basis \(\{e_k\}_1^\infty\) for which this inequality holds will be called an \(\{r,s\}\)-basis. By the pair of numbers \(\{r,s\}\) one can classify bases in \(E\). Thus, the class of all quasi-normalized unconditional\(^*\) bases in Hilbert space \(H\), by the theorem of Gelfand \((^3)\), coincides with the class of all \(\{2,2\}\)-bases.

The exact least upper (greatest lower) bound of the numbers \(r\) (respectively \(s\)) for which inequality (3) holds will be called the lower (upper) degree of the basis \(\{e_k\}_1^\infty\) and denoted respectively by \(\rho=\rho(\{e_k\}_1^\infty)\), \(\sigma=\sigma(\{e_k\}_1^\infty)\). If the least upper or greatest lower bound is not attained, then we shall use parentheses, applying the term \((\rho,\sigma)\)-basis, and if

\(^*\) A basis \(\{e_k\}_1^\infty\) is called unconditional if it remains a basis under any permutation of its elements. A basis that is not unconditional is called conditional.

is attained—square brackets. Different brackets, for example \([\rho,\sigma)\) or \((\rho,\sigma]\), are understood in the natural way. If \(\rho(\{e_k\}_1^\infty)\le r,\ \sigma(\{e_k\}_1^\infty)\ge s,\ 1<r\le s<\infty\), then we shall call \(\{e_k\}_1^\infty\) an \(\langle r,s\rangle\)-basis.

The main result of this section is the non-improvability of inequality (3) for the case of a separable Hilbert space \(H\), in the sense that for arbitrary numbers \(r\) and \(s\), \(1<r\le s<\infty\), there exists in \(H\) an \(\langle r,s\rangle\)-basis. An essential role in this is played by the systems

\[ \{|t|^\alpha \cos nt\}_{-\infty}^{\infty},\qquad \{|t|^{-\alpha}\cos nt\}_{-\infty}^{\infty},\qquad 0<\alpha<{}^1\!/_{2}, \tag{4} \]

which, as follows from the results of K. I. Babenko \((^2)\), are conditional bases in the closure of their linear span in \(L_2[-\pi,\pi]\). Along the way, a number of theorems are obtained on the coefficients of expansions with respect to bases in arbitrary Banach spaces.

Theorem 3. Let \(\{e_k\}_1^\infty\) be a quasinormalized sequence in a Banach space \(E\) satisfying the conditions:

1. \[ \left\|\sum_1^n e_k\right\|\ge Kn^r\qquad (K>0,\ r>0),\qquad n=1,2,\ldots . \]

2. For any finite sets of numbers \(\{\alpha_i\}_1^n,\ \{\beta_i\}_1^n\), from the condition \(0\le \alpha_k\le \beta_k,\ k=1,2,\ldots,n\), it follows that

\[ \left\|\sum_1^n \alpha_k e_k\right\|\le \left\|\sum_1^n \beta_k e_k\right\|. \]

If \(\{\alpha_k\}_1^\infty\) is a positive monotonically decreasing sequence such that \(\{\alpha_k\}_1^\infty\in l_p\), where \(p>1/r\), then the series \(\sum_1^\infty \alpha_k e_k\) diverges.

Theorem 4. If a quasinormalized sequence \(\{e_k\}_1^\infty\) in a Banach space \(E\), for some \(r\) and \(K\), \(0<r\le 1,\ 0<K<\infty\), satisfies the condition

\[ \left\|\sum_1^n e_k\right\|\le Kn^r,\qquad n=1,2,\ldots, \]

and the positive sequence \(\{\alpha_k\}_1^\infty\downarrow 0\) belongs to \(l_p\), where \(0<p<1/r\), then the series

\[ \sum_1^\infty \alpha_k e_k \]

converges.

With the aid of several lemmas it is verified that the bases (4) satisfy the conditions of Theorems 3 and 4 (with the corresponding choice of the parameter \(\alpha\)). Using these considerations, the following Theorems 5—7 are established.

Theorem 5. Let \(\{\alpha_k\}_1^\infty\) be a numerical sequence such that \(|\alpha_1|\ge |\alpha_2|\ge\cdots\). In order that, for every quasinormalized basis \(\{e_k\}_1^\infty\) in \(H\), the series

\[ \sum_1^\infty \alpha_k e_k \]

converge, it is necessary and sufficient that \(\{\alpha_k\}_1^\infty\in l_p\) for every \(p>1\).

Theorem 6. Let \(\{\alpha_k\}_1^\infty\) be a numerical sequence such that \(|\alpha_1|\ge |\alpha_2|\ge\cdots\). In order that in \(H\) there exist a quasinormalized basis \(\{e_k\}_1^\infty\) such that the series

\[ \sum_1^\infty \alpha_k e_k \]

converges, it is necessary and sufficient that \(\{\alpha_k\}_1^\infty\in l_p\) for some \(p,\ 1<p<\infty\).

Theorem 7. For arbitrary numbers \(r,s,\ 1<r\le s<\infty\), in a separable Hilbert space \(H\) there exists an \(\langle r,s\rangle\)-basis.

Theorem 7 generalizes M. Sh. Altman’s result on the existence, in a separable Hilbert space, of a basis that is neither Hilbertian nor Besselian \((^1)\). As an application of Theorem 7 we obtain

a negative solution to the question of the existence in a Hilbert space \(H\) of a universal basis \(\{e_k\}_1^\infty\), i.e., one such that every normalized basis \(\{g_i\}_1^\infty\) in \(H\) is equivalent* to some subsequence \(\{e_{k_i}\}_1^\infty\) of the basis \(\{e_k\}_1^\infty\).

Theorem 8. In a separable Hilbert space \(H\) there does not exist a universal basis.

Kharkov Polytechnic Institute
named after V. I. Lenin

Received
15 I 1970

REFERENCES

¹ M. Sh. Altman, DAN, 69, 483 (1949).
² K. I. Babenko, DAN, 62, 157 (1948).
³ I. M. Gelfand, Uch. zap. Moscow Univ., 148, Mathematics 4, 224 (1951).
⁴ M. G. Krein, D. P. Milman, and M. A. Rutman, Zap. Kharkov Math. Soc., 16, 14 (1940).
⁵ J. Lindenstrauss, Michigan Math. J., 10, 241 (1963).
⁶ L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, “Nauka,” 1965.

* Two bases \(\{e_k\}_1^\infty\) and \(\{g_k\}_1^\infty\), respectively in Banach spaces \(E\) and \(G\), are called equivalent if there exists an isomorphism \(T\) of \(E\) onto \(G\) such that \(Te_k = g_k,\ k = 1, 2, \ldots\).