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.
Contents
- Definitions
- An elementary example cocompactness for displaystyle ell infty hookrightarrow ell infty
- Some known embeddings that are cocompact but not compact
- References
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.
Definitions
Let
A continuous embedding of two normed vector spaces,
An elementary example: cocompactness for ℓ ∞ ↪ ℓ ∞ {\displaystyle \ell ^{\infty }\hookrightarrow \ell ^{\infty }}
Embedding of the space