Applications of functional data with large numbers of predictors have grown precipitously in recent years, driven, in part, by rapid advances in genotyping technologies. Given the large numbers of genetic mutations encountered in genetic association studies, statistical methods which more fully exploit the underlying structure of the data are imperative for maximizing statistical power. However, there is currently very limited work in functional data with large numbers of predictors. Tools are presented for simultaneous variable selection and parameter estimation in a functional linear model with a functional outcome and a large number of scalar predictors; the technique is called AFSL for $\textit{Adaptive Function-on-Scalar Lasso}.$ It is demonstrated how techniques from convex analysis over Hilbert spaces can be used to establish a functional version of the oracle property for AFSL over any real separable Hilbert space, even when the number of predictors, $I$, is exponentially large compared to the sample size, $N$. AFSL is illustrated via a simulation study and data from the Childhood Asthma Management Program, CAMP, selecting those genetic mutations which are important for lung growth.