### A generalized Kahane-Khinchin inequality

The inequality $\u0283log|\sum {a}_{n}{e}^{2\pi i{\phi}_{n}}|d{\phi}_{1}\dots d{\phi}_{n}\ge Clog\left(\sum \right|{a}_{n}{{|}^{2})}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of ${a}_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by ${e}^{2\pi i\phi}$.