Optimal unbounded control problems with linear growth w.r.t. the control, both in the dynamics and in the cost, may fail to have minimizers in the class of absolutely continuous state trajectories. For this reason, extended versions of such problems have been investigated, in which the domain is extended to include possibly discontinuous state trajectories of bounded variation, and for which existence of minimizers is guaranteed. It is of interest to know whether the passage from the original optimal control problem to its extension introduces an infimum gap. This will reveal whether it is possible to approximate extended minimizers by absolutely continuous state trajectories, as might be required for engineering implementation, and whether numerical schemes might be ill-conditioned. This paper provides sufficient conditions under which there is no infimum gap, expressed in terms of normality of extremals. The link we establish between infimum gaps and normality gives insights into the infimum gap phenomenon. But, perhaps more importantly, it opens up a new approach to devising useful tests for the absence of infimum gaps, namely to supply verifiable sufficient conditions for normality of extremals. We give several examples of the use of this approach, and show that it leads to either new conditions, or improvement of known conditions, for no infimum gaps. We also give a criterion for non infimum gaps, which covers some problems where the normality condition is violated, illustrating that sufficient conditions of normality type, while covering many cases, are not necessary. ✩