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 { mathfrak A} be a C^* -algebra, T be a locally compact Hausdorff space equipped with a Radon measure mu and let (A_t)_{t in T} be a continuous field of operators in { mathfrak A} such that the function t mapsto A_t$ is norm continuous on T and the function t mapsto A_t is integrable. Let alpha: T times T to mathbb{C} be a measurable function such that overline { alpha(t,s)} alpha(s,t)=1 for all t, s in T . Then begin{align*} int_T int_T& big| alpha(t,s) A_t- alpha(s,t) A_s big|^2d mu(t)d mu(s)+ int_T int_T big| alpha(t,s) B_t- alpha(s,t) B_s big|^2d mu(t)d mu(s) nonumber &= 2 int_T int_T big| alpha(t,s) A_t- alpha(s,t) B_s big|^2d mu(t)d mu(s) - 2 big| int_T(A_t-B_t)d mu(t) big|^2 ,. end{align*}

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