UDC 513.88:513.83

MATHEMATICS

V. I. GURARII

ON SUBSPACES AND BASES IN SPACES OF CONTINUOUS FUNCTIONS

(Presented by Academician L. V. Kantorovich on 3 VII 1965)

In the present note a number of theorems are given on subspaces and bases in spaces \(C(T)\)*, where \(T\) is a metric compactum, and also in Banach subspaces of a certain abstract class that includes the class of all spaces of the form \(C(T)\). In this connection, the case of a subspace of \(C\) will be considered in greater detail. All spaces considered here are assumed to be real.

§ 1. We shall use the following terminology and notation.

1. Let \(\{x_i\}_{i=1}^{\infty}\) be a sequence of pairwise distinct elements of a metric compactum \(T\). A basis \(\{f_i\}_{i=1}^{\infty}\) in \(C(T)\) is called interpolating with nodes \(\{x_i\}_{i=1}^{\infty}\) if, in the expansion of any \(f \in C(T)\),

\[ f=\sum_{i=1}^{\infty} a_i f_i, \]

the \(n\)-th partial sum \(\sum_{i=1}^{n} a_i f_i\) interpolates \(f(x)\) at the nodes \(x_1,\ldots,x_n\).

1. We shall denote by \(\gamma_{\{e_i\}_{i=1}^{\infty}}\) the index of the sequence \(\{e_i\}_{i=1}^{\infty}\) of elements of a Banach space (see the definition in (2) or (3)).

2. For isomorphic Banach spaces \(E_1\) and \(E_2\), we shall denote by \(d(E_1,E_2)\) the quantity

\[ d(E_1,E_2)=\inf_T \|T\|\cdot\|T^{-1}\|, \]

where \(T\) ranges over all isomorphisms of \(E_1\) onto \(E_2\) (the quantity \(\ln d(E_1,E_2)\) was introduced by S. Banach and S. Mazur (see (4), p. 211)).

1. For a subspace \(P\) of a Banach space \(E\), adopt the notation

\[ \lambda(P,E)=\inf_A \|A\|, \]

where \(A\) ranges over all projection operators from \(E\) onto \(P\). The projection constant of a Banach space \(R\) is the quantity

\[ \lambda(R)=\sup_B \lambda(R,B), \]

where \(B\) ranges over all Banach spaces containing \(R\) as a subspace.

Theorem 1. Let \(\{x_i\}_{i=1}^{\infty}\) be a sequence of pairwise distinct elements of a metric compactum \(T\). In order that there exist in \(C(T)\) an

* \(C(T)\) (\(T\) a metric compactum) is the space of all continuous functions on \(T\), with the naturally defined vector operations and the norm \(\|f\|=\max |f(x)|\). If \(T\) is a collection of \(n\) points, then \(C(T)\) is denoted by \(c^n\), \(n=1,2,\ldots\).

interpolating basis with nodes \(\{x_i\}_{i=1}^{\infty}\), it is necessary and sufficient that \(\{x_i\}_{i=1}^{\infty}\) be dense in \(T\).

Theorem 2. Let \(\{x_i\}_{i=1}^{\infty}\) be a sequence of pairwise distinct elements of the metric compactum \(T\), dense in \(T\). For any preassigned \(\varepsilon>0\), there exists in \(C(T)\) an interpolating basis \(\{e_i\}_{i=1}^{\infty}\) with nodes \(\{x_i\}_{i=1}^{\infty}\) such that:

1) \(\gamma_{\{e_i\}_{i=1}^{\infty}}<1-\varepsilon\);

2) \(d(L_{1,n},c^n)<1+\varepsilon,\quad n=1,2,\ldots,\) where \(L_{1,n}\) is the linear span of \(e_1,\ldots,e_n\).

Define the class \(\mathfrak C\) of Banach spaces, assuming that \(E\in\mathfrak C\) if, for every \(\varepsilon>0\) and every finite-dimensional subspace \(P\) of \(E\), there exists a finite-dimensional subspace \(Q=Q(P,\varepsilon)\) in \(E\), \(Q\supset P\), such that \(d(Q,c^n)<1+\varepsilon\), where \(n=\dim Q\).

Theorem 3. If in a Banach space \(E\), for arbitrary \(\varepsilon>0\) and finite-dimensional subspace \(P\), there exists a finite-dimensional subspace \(Q=Q(P,\varepsilon)\) such that the conditions

1) \(d(Q,c^n)<1+\varepsilon\) \((n=\dim Q)\);

2) \(\displaystyle \sup_{x\in P,\ \|x\|=1}\inf_{y\in Q}\|x+y\|<\varepsilon,\)

are satisfied, then \(E\in\mathfrak C\).

It follows from Theorem 3, for example, that \(c_0\) is a space of class \(\mathfrak C\), since the linear span of the first \(n\) elements of the natural basis in \(c_0\) is isometric to \(c^n\) (for the definition of the spaces \(c_0\) and \(c\), see (⁴)).

Theorem 4. For any metric compactum \(T\), \(C(T)\in\mathfrak C\).

Theorem 5. For any metric compactum \(T\), \(d(c_0,C(T))\geqslant 2\).

Theorem 5 means that \(d(c_0,c)\geqslant 2\) and thereby, in particular, gives an answer to one question of S. Banach ((⁴), p. 211). Theorems 4 and 5 mean, moreover, that the class of all spaces of the form \(C(T)\) is a proper subset of the set of all separable spaces of class \(\mathfrak C\).

For an arbitrary set \(\mathfrak S\), denote by \(m(\mathfrak S)\) the space of all real bounded functions on \(\mathfrak S\), with the naturally defined vector operations and the norm
\[ \|f\|=\sup_{x\in\mathfrak S}|f(x)|. \]

Theorem 6. For any set \(\mathfrak S\), \(m(\mathfrak S)\in\mathfrak C\).

Theorem 7. In a separable space of class \(\mathfrak C\) there exists a basis.

Each of Theorems 1, 2, and 7 is a generalization of a theorem of F. S. Vakher on the existence of a basis in the space \(C(T)\), where \(T\) is a metric compactum (⁵).

Theorem 8. If \(E\) is an infinite-dimensional space of class \(\mathfrak C\), then in \(E\) there exists an infinite-dimensional subspace \(E'\), not isomorphic to \(E\)*.

Theorem 9. If \(P\) is a finite-dimensional subspace of a space \(E\) of class \(\mathfrak C\), then \(\lambda(P,E)=\lambda(P)\).

§ 2. We present two theorems on subspaces in \(C\), which consist of functions possessing certain differentiability properties (some results on basic sequences in \(C\) consisting of such functions were obtained in (⁶)**).

Theorem 10. If all elements of a subspace \(E\) of the space \(C\) are functions differentiable on \([0,1]\), then \(E\) is finite-dimensional.

Nevertheless, in \(C\) there exist subspaces of infinite dimension consisting of functions differentiable on \((0,1)\) (and even analytic on \((0,1)\)). An example may be furnished by the closure in \(C\) of the linear

* All subspaces considered here are assumed to be closed.

* A sequence \(\{e_i\}_{i=1}^{\infty}\) in a Banach space is called basic* if it is a basis in the closure of its linear span.

envelopes of the sequence of powers \(\{t^{n_k}\}_{k=1}^{\infty}\), \(n_k>0\),

\[ \sum_{k=1}^{\infty}\frac{1}{n_k}<\infty \tag{7} \]

However, the class of such subspaces in \(C\) is very narrow, as the following shows.

Theorem 11. If all elements of an infinite-dimensional subspace \(E\) of the space \(C\) are functions differentiable on \((0,1)\), then for every \(\varepsilon>0\) there exists a subspace \(E_\varepsilon\) in \(E\) such that \(d(E_\varepsilon,c_0)<1+\varepsilon\).

Corollary. If the elements of a reflexive subspace \(E\) of the space \(C\) are functions differentiable on \((0,1)\), then \(E\) is finite-dimensional.

Kharkov
Automobile and Highway Institute

Received
29 VI 1965

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