Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a càdlàg random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^θ\setminus K^{-θ})$ that the target process is close to the boundary of $K$.

Ver 4: 33 pages, added Example 1.10, improved some details and discussion, and streamlined some notation; Ver3: 28 pages, corrected a mistake in Lemma 1.10 and added discussion and details, only superficial changes to the main result; Ver2: 21 pages, additional discussion and details; Ver1: 17 pages