A uniformly bounded complete Euclidean system

Abstract

A uniformly bounded complete orthonormal system of functions $\Theta =\{{\theta }_n{\}}_{n=1}^{\infty },$ $\Vert {\theta }_n{\Vert }_{L_{[0,1]}^{\infty }}\le M$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0,1]$ if $\{a_n{\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a. e. for any $\{a_n{\}}_{n=1}^{\infty }\notin l^2$. Thus Menshov's theorem on the representation of measurable, almost everywhere finite, functions by almost everywhere convergent trigonometric series cannot be extended to the class of uniformly bounded complete orthonormal systems.