We give a general version of Bryc's theorem valid on any topological space and with any algebra A of real-valued continuous functions separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and A constituted by functions vanishing at infinity, we give a sufficient condition on the functional Lambda(.)|(A) to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are genealized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on Lambda(.)|(A).