We investigate the L-2 gain of periodic linear switched systems under fast switching. For systems that possess a suitable notion of a time-average system, we characterize the relationship between the L-2 gain of the switched system and the L-2 gain of its induced time- average system when the switching rate is sufficiently fast. We show that the switched system L-2 gain is in general different from the average system L-2 gain if the input or output coefficient matrix switches. If only the state coefficient matrix switches, the input- output energy gain for a fixed L-2 input signal is bounded by the L-2 gain of the average system as the switching rate grows large. Additionally, for a fixed L-2 input, the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows.