L-p convergence of a lacunary series is equivalent to ell-two coefficients

If $(a_j)\in {\mathrm{\ell }}^2$, its coefficient tails tend to zero. Apply the upper bound in [F1] to each difference $S_m-S_n$ of partial sums. The differences tend to zero in the $L^p$ norm for $p\ge 1$ and in the metric $d_p$ for $0<p<1$, since $d_p(S_m,S_n)=\mathrm{\parallel }{S}_m-S_n{\mathrm{\parallel }}_p^p$.

By [F2], these Cauchy sequences have limits in their respective spaces.

Conversely, convergence makes the partial sums Cauchy in the relevant norm or metric. The lower bound in [F1] applied to every $S_m-S_n$ forces ${\sum }_{n<j\le m}\mathrm{\mid }{a}_j{\mathrm{\mid }}^2\to 0$ uniformly as $m>n\to \mathrm{\infty }$; in the metric case take the $p$-th root first. The nonnegative partial sums ${\sum }_{j\le n}\mathrm{\mid }{a}_j{\mathrm{\mid }}^2$ are then Cauchy and converge, so $(a_j)\in {\mathrm{\ell }}^2$. This proves both implications.