MATHEMATICS

V. I. GURARII

ON BASES IN SPACES OF CONTINUOUS FUNCTIONS

(Presented by Academician V. I. Smirnov on 21 VII 1962)

§ 1. Let \(R\) and \(S\) be sets in a Banach space \(E\); let \(P\) and \(Q\) be the closures of their linear spans. We shall call the inclination of \(R\) to \(S\) the quantity

\[ (\widehat{R;S})=\inf_{x\in P;\ \|x\|=1}\rho(x;Q). \]

For a sequence \(\{e_i\}_{i=1}^{\infty}\), \(e_i\in E\), \(i=1,2,\ldots\), denote by \(P_{ij}\) the linear span of the elements \(e_i,e_{i+1},\ldots,e_j\) \((i\le j)\), and we shall call the index of \(\{e_i\}_{i=1}^{\infty}\) the quantity

\[ \gamma_{\{e_i\}}=\inf_{n;\ m;\ n<m}(\widehat{P_{1;n};P_{n+1;m}}). \]

Obviously, \(0\le \gamma_{\{e_i\}}\le 1\). If \(\gamma_{\{e_i\}}=1\), then \(\{e_i\}_{i=1}^{\infty}\) is called orthogonal. As M. M. Grinblum showed \((^1)\), in order that the sequence \(\{e_i\}_{i=1}^{\infty}\) be a basis* in \(E\), it is necessary and sufficient that \(\gamma_{\{e_i\}}>0\).

We adopt the notation: \(C\) is the space of all functions continuous on \([0,1]\) with norm \(\|\varphi\|=\max_{0\le t\le 1}|\varphi(t)|\), and \(\widetilde C\) is the space of all real-valued functions continuous on \([0,1]\) with the same norm.

Theorem 1. If a sequence of functions differentiable on \((0,1)\), \(\{e_i(t)\}_{i=1}^{\infty}\), in \(\widetilde C\) satisfies the conditions:

1) the set of stationary points of any linear combination

\[ \sum_{i=1}^{n}\alpha_i e_i(t) \]

has no limit points in \((0,1)\);

2) in the closure of the linear span of \(\{e_i\}_{i=1}^{\infty}\) there exists a function which, together with its derivative, preserves its sign on \((0,1)\), then \(\gamma_{\{e_i\}}<1\).

Denote by \(M_f\) the set of points of greatest value on \([0,1]\) of the function \(|f(t)|\).

Lemma 1. For a given function \(f(t)\in\widetilde C\) and \(\varepsilon>0\) there exists \(\delta>0\) such that for any function \(g(t)\in\widetilde C\), \(\|g\|<\delta\):

\[ M_{f+g}\subset \bigcup_{t\in M_f}(t-\varepsilon,\ t+\varepsilon). \]

* In works by foreign authors such a sequence is sometimes called monotone, apparently proceeding from the fact that

\[ \left\|\sum_{i=1}^{m}\alpha_i e_i\right\|\ge \left\|\sum_{i=1}^{n}\alpha_i e_i\right\| \quad \text{for } m\ge n. \]

** The sequence \(\{e_i\}_{i=1}^{\infty}\) is called a basis in \(E\) if it is a basis in the closure of the linear span over it.

Lemma 2. Let \(t'\), \(t''\) be respectively the least and the greatest of the values of \(t\) at which the maximum value of the modulus of the function \(f(t)\), differentiable on \((0,1)\), is attained, \(0<t'<t''<1\), and for some function \(g(t)\) let, on the interval \((a,b)\supset M_f\), \(g(t)>0\) and \(g'(t)\) preserve its sign. Then there exists \(\alpha_0>0\) such that, for \(0<\alpha<\alpha_0\),

\[ M_{f+\alpha g}\subset (t'',b), \quad \text{if } g'(t)>0 \text{ and } f(t'')>0, \]

\[ M_{f+\alpha g}\subset (a,t'), \quad \text{if } g'(t)<0 \text{ and } f(t')>0. \]

Lemma 3. In order that \((f;g)<1\), \(f\in\widetilde C\), \(g\in\widetilde C\), it is necessary and sufficient that one of the following conditions be fulfilled:

\[ \begin{aligned} &1)\quad \operatorname{sign} f(t)=\operatorname{sign} g(t) \quad \text{for } t\in M_f,\\ &2)\quad \operatorname{sign} f(t)=-\operatorname{sign} g(t) \quad \text{for } t\in M_f. \end{aligned} \]

Let us outline the proof of Theorem 1. There is a function \(f_1(t)\in P_{1;3}\), \(f_1(0)=f_1(1)=0\), and if \(t_0\) is the greatest of the values of \(t\) for which \(|f_1(t)|\) has its maximum value on \([0,1]\), then \(0<t_0<1\). For definiteness assume that \(f_1(t_0)>0\), and for the function \(g(t)\) defined in condition 2 of the theorem we have \(g(t)>0\), \(g'(t)>0\), \(0<t<1\).

It follows from condition 1 that on some interval \((t_0,\tau)\), \(f'_1(t)<0\). From Lemmas 1 and 2 it follows that for some natural number \(n\) there is a function \(f_2(t)\in P_{1;n}\) such that \(M_{f_2}\subset (t_0,\tau)\) and \(f_2(t'_0)>0\), where \(t'_0\) is the least of the points of maximum value of \(|f_2(t)|\) on \([0,1]\); \(t_0<t'_0<\tau\). Consider any function \(h(t)\in P_{1;N}\), where \(N\) is arbitrary. On the basis of condition 1 and Rolle’s theorem, \(h(t)\) preserves its sign on some interval \((\tau_0,t'_0)\), \(t_0<\tau_0<t'_0\). By Lemma 2 we obtain that, for some \(\alpha>0\), the function \(f_3(t)=f_2(t)+\alpha f_1(t)\) satisfies the conditions:

\[ M_{f_3}\subset (\tau_0,t'_0) \quad \text{and} \quad f_3(t)>0 \text{ for } t\in M_{f_3}. \]

But then, by Lemma 3, \((f_3;h)<1\), and, since \(f_3(t)\in P_{1;n}\), it follows that \((P_{1;n};h)<1\). Hence \(\gamma_{\{e_i\}}<1\).

Corollary. In the space \(\widetilde C\) there does not exist an orthogonal basis consisting of functions analytic on \((0,1)\).

This, in particular, means the impossibility of constructing in \(\widetilde C\) an orthogonal basis consisting of polynomials or trigonometric polynomials.

We note that in \(\widetilde C\) there exist bases of polynomials that are orthogonal in the metric \(L^2\) \((^2)\).

With the aid of Theorem 1 one establishes

Theorem 2. If the elements of the space \(E\subset\widetilde C\) are functions analytic on \((0,1)\) and, moreover, there exists a function \(g(t)\in E\) which preserves its sign on \((0,1)\) together with its first derivative, then there is no orthogonal basis in \(E\).

Examples of spaces satisfying the conditions of Theorem 2 may be the closures in \(\widetilde C\) of the linear spans of the powers \(\{t^{n_k}\}_{k=1}^{\infty}\),

\[ n_k>0,\quad \sum_{k=1}^{\infty}\frac{1}{n_k}<\infty \quad (^{3}). \]

Thus the question of the existence of infinite-dimensional separable Banach spaces not having an orthogonal basis is answered.

§ 2. A subspace \(E\subset C\) will be called saturated if, whatever the partition

\[ [0,1)=\bigcup_{i=0}^{n-1}[a_i,a_{i+1}), \qquad 0=a_0<a_1<\cdots \]

… \(<a_n=1\), and the function \(f(t)\in C\), there exist functions \(g(t)\in E\) and a collection of points \(\{t_i\}_{i=1}^n\), \(t_i\in(a_{i-1}a_i)\), \(i=1,2,\ldots\), such that \(f(t_i)=g(t_i)\). With the aid of the Banach–Mazur theorem \((^4)\) one can obtain

Theorem 3. Every infinite-dimensional separable Banach space is isomorphic to a saturated subspace of the space \(C\).

Definition. Let \(\mathfrak M\) be some class of subsequences of the sequence of natural numbers; a sequence \(\{e_i\}_{i=1}^{\infty}\) will be called a basis in \(E\) relative to \(\mathfrak M\) if it is complete in \(E\) and every subsequence \(\{e_{n_k}\}_{k=1}^{\infty}\) for which \(\{n_k\}_{k=1}^{\infty}\in\mathfrak M\) is a basis in \(E\).

Example 1. The sequence \(\{s_i(t)\}_{i=1}^{\infty}\), where \(s_{2k-1}=\cos(k-1)t\), \(s_{2k}=\sin kt\), \(k=1,2,\ldots\), is a basis in \(C\) relative to the class of all lacunary subsequences of the sequence of natural numbers \((^5)\).

Example 2. The sequence of powers \(\{t^k\}_{k=1}^{\infty}\) is a basis in \(C\) relative to the class of all subsequences of the sequence of natural numbers satisfying the condition:
\[ \lim_{k\to\infty}\frac{n_{k+1}}{n_k}>A, \]
where \(A\) is an absolute constant.

The last assertion is a special case of the following theorem.

Theorem 4. In order that the sequence of powers \(\{t^{n_k}\}_{k=1}^{\infty}\), where \(\{n_k\}_{k=1}^{\infty}\) is a positive increasing sequence, be a basis in \(C\), it is necessary that
\[ \lim_{k\to\infty}\frac{n_{k+1}}{n_k}>1, \]
and it is sufficient that
\[ \lim_{k\to\infty}\frac{n_{k+1}}{n_k}>A, \]
where \(A\) is an absolute constant.

Let \(\{p_k\}_{k=1}^{\infty}\) be a subsequence of the sequence of natural numbers. Denote by \(\mathfrak M_{\{p_k\}}\) the class of all subsequences of the sequence of natural numbers each of which does not contain at least one number from each pair of the form \(\{p_k,\ p_k+1\}\), \(k=1,2,\ldots\).

Theorem 5. In an infinite-dimensional separable Banach space, for any subsequence \(\{p_i\}_{i=1}^{\infty}\) of the sequence of natural numbers there exists a basis relative to \(\mathfrak M_{\{p_i\}}\).

The idea of the proof consists in constructing a basis relative to \(\mathfrak M_{\{p_i\}}\) in an arbitrary saturated subspace of the space \(C\), followed by an application of Theorem 3.

Putting \(p_i=i\), \(i=1,2,\ldots\), in Theorem 5, we obtain

Corollary. In an infinite-dimensional separable Banach space \(E\) there exists a complete sequence \(\{e_i\}_{i=1}^{\infty}\) such that any subsequence \(\{e_{n_i}\}_{i=1}^{\infty}\) for which \(n_{i+1}-n_i>1\), \(i=1,2,\ldots\), is a basis in \(E\).

Received
16 VII 1962

REFERENCES

\(^1\) M. M. Grinblyum, DAN, 31, No. 5, 428 (1941).
\(^2\) K. M. Shaidukov, Scientific Transactions of the Kazan Institute of Civil Engineering, Petroleum Industry, 5, 119 (1957).
\(^3\) A. F. Leont’ev, DAN, 72, No. 4, 621 (1950).
\(^4\) S. Banach, Course of Functional Analysis, 1948, pp. 157–159.
\(^5\) N. K. Bari, Trigonometric Series, Moscow, 1961, pp. 178–179.