The estimation of a system’s infinity norm using one set of measured input and output data is investigated. It is known that, if the data set is noise free, this problem can be solved using convex optimization. In the presence of noise, convergence of this estimate to the true infinity norm of the system is no longer guaranteed. In this paper, a convex noise set is defined in the time domain using decorrelation between the noise and the system input. For infinite data length, we prove that the estimate of the infinity norm converges to its true value. A simulation example shows the behavior for finite data length. In addition, the method is used to test closed-loop stability in the context of data-driven controller tuning. A sufficient condition for stability in terms of an infinity norm is introduced. The effectiveness of the proposed stability test is illustrated via a simulation example.