Bases in the spaces $C$ and $L_p$

In this paper it is proved that for any numbers $A$ and $B$, $0<A<B$, there exists a basis in the space $C$ consisting of functions $\varphi_k(x)$, $k=1,2,\dots$, whose graphs lie in the strip $0\leqslant x\leqslant1$, $A\leqslant y\leqslant B$. It is shown that for the space $L_p$, $p>1$, there is no analogous “basis in a strip” theorem.