For time-invariant finite-dimensional systems, it is known that global asymptotic sta- bility (GAS) is equivalent to uniform GAS (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e., under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or H\"older spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equiv- alences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces.