If embeds compactly into and embeds continuously into , then boundedness in together with a suitable time-derivative bound in makes a family relatively compact in . A standard case is
If and for a dense continuous embedding and some , then has a representative in . Indeed, its pairing with each element of a dense subset of is absolutely continuous, and the uniform bound extends continuity to every element of .
If is weakly continuous and an energy equality makes continuous, then is strongly continuous. This follows from the Radon-Riesz theorem applied whenever .
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Aubin–Lions lemma is a result in the field of functional analysis, particularly in the study of the convergence of sequences of functions, and is often used in the context of nonlinear partial differential equations. The lemma provides conditions under which compactness can be guaranteed for a sequence of functions in certain function spaces. More specifically, it deals with the convergence properties of families of bounded sets in reflexive Banach spaces.