  We reprove an inequality for Rademacher series with coefficients in the
Schatten class $S_1$. Our method yields the same estimate for coefficients
after suitable gaps in $S_1$-valued trigonometric series; this was known for
scalar-valued functions. A very similar method gives a new proof of the
extension to $S_1$-valued $H^1$ functions of Paley's theorem about lacunary
coefficients.
