Simple proofs of Olevskij type theorems (Q1358483)

In 1960, P. L. Ul'yanov conjectured that an orthonormal system of functions in \(C([0;1])\) cannot be a basis of that space if this system consists of uniformly bounded functions. In 1965, A. M. Olevskii proved Ul'yanov's conjecture on the basis of the following theorem, also proved by Olevskii. Let \((e_n)\) be an orthonormal system of uniformly bounded functions, and let \((c_n)\) be a numerical sequence satisfying the following condition: \[ \varlimsup_{n\to\infty} {1\over n} \sum^n_{k= 1} c^2_k> 0. \] Then \[ \varlimsup_{n\to\infty} \int^1_0 \Biggl|\sum^n_{k= 1} c_k e_k(t)\Biggr|dt= \pm\infty. \] The natural question may be raised as to whether and to what extent Olevskii's theorem can be generalized to a wider class of separable Banach spaces. In this paper, the first step towards the solution of this problem is made; namely, Olevskii's theorem and one of its generalizations are proved by methods of functional analysis.