Cocompact embedding

In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name (Lemma 6),(Lemma 2.5),(Theorem 1), or by ad-hoc monikers such as vanishing lemma or inverse embedding.

Cocompactness property allows to verify convergence of sequences, based on translational or scaling invariance in the problem, and is usually considered in the context of Sobolev spaces. The term cocompact embedding is inspired by the notion of cocompact topological space.

Let ${\displaystyle G}$ be a group of isometries on a normed vector space ${\displaystyle X}$. One says that a sequence ${\displaystyle (x_{k})\subset X}$ converges to ${\displaystyle x\in X}$ ${\displaystyle G}$-weakly, if for every sequence ${\displaystyle (g_{k})\subset G}$, the sequence ${\displaystyle g_{k}(x_{k}-x)}$ is weakly convergent to zero.

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