On a problem of S. Banach from The Scottish book

Abstract:

Denote by $B\left ( \Phi \right )$ the closure in ${L^2}$ of linear combinations of functions of the orthonormal system $\Phi = \left \{ {{\Phi _n}} \right \}_{n = 1}^\infty$. Denote by $B{(\Phi )^ \bot }$ the orthogonal complement of $B(\Phi )$ in ${L^2}$. We prove the following theorem: Let $\varphi$ be a nonnegative measurable function defined on $\left [ {0, + \infty } \right )$ such that ${\text {li}}{{\text {m}}_{x \to + \infty }}{x^2}/\varphi \left ( x \right ) = 0$ and $N$ is a positive integer or $N = + \infty$. Then there is a uniformly bounded orthonormal system ${\Phi ^{\left ( N \right )}} = \left \{ {{\Phi _n}} \right \}_{n = 1}^\infty$ such that $\operatorname {dim} B{({\Phi ^{(N)}})^ \bot } = N$ and, for every nontrivial function $f$ from $B{({\Phi ^{(N)}})^ \bot },\int {\varphi \left ( {\left | {f\left ( t \right )} \right |} \right )} dt = + \infty$.