A general operator parallelogram law concerning a characterization of inner product spaces is established and an operator extension of Bohr s inequality as well as several norm inequalities are presented. More precisely, let A be a C -algebra, T be a locally compact Hausdor space equipped with a Radon measure and let (At)t2T be a con- tinuous eld of operators in A such that the function t 7! At is norm continuous on T and the function t 7! kAtk is integrable. Let  : T T ! C be a measurable function such that (t; s)(s; t) = 1 for all t; s 2 T. Then ZT ZT (t; s)At

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