This article is cited in 2 papers

Approximation properties of finite-dimensional subspaces in $L_1$

S. Ya. Havinson, Z. S. Romanova


Abstract: It is known that if a measure $\mu$ has no atoms, then the space $L_1(T,\mu)$ contains no finite-dimensional Chebyshev subspace. In the present work it is shown that an arbitrary finite-dimensional subspace $E$ in $L_1(T,\mu)$ (for which the measure has no atoms) is almost Chebyshev, i.e. the set of elements possessing nonunique best approximations in the given finite-dimensional space $E$ is of the first category. At the same time this set is everywhere dense. There is further given a characterization of elements with nonunique best approximations.
Bibliography: 16 titles.

UDC: 517.512+519.56

MSC: Primary 41A50; Secondary 46E30

Received: 03.06.1971

 English version: Mathematics of the USSR-Sbornik, 1972, 18:1, 1–14

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