Bohr compactification

Summary

The Bohr compactification of a topological group $G$ is the largest compact Hausdorff group into which $G$ maps continuously. Every continuous homomorphism from $G$ to a compact Hausdorff group factors through it in a unique way. For a locally compact abelian group it is the dual of the dual group made discrete. The continuous functions on it correspond to the almost periodic functions on $G$.