We consider the inviscid surface quasigeostrophic (SQG) equation and develop a weak formulation for its solutions. Starting from smooth, non-decaying solutions of the dissipative SQG equation, we establish an inviscid limit. Using an a prior estimate for u in the uniformly local L^2 space, we prove convergence for initial data \theta_0 in L^\infty and u_0 in uniformly local L^2 satisfying the compatibility condition that u equals the perpendicular of the Riesz transform of \theta on the homogenous Littlewood-Paley blocks. The limiting inviscid solution is shown to exist globally in time and to remain bounded in the same functional spaces as its initial data. Finally, we verify that this inviscid limit satisfies the definition of a weak solution, and with sufficient regularity, becomes a classical solution.