We present a translation of the asynchronous pi-calculus into linear concurrent constraint languages (LCC), and use that translation and the recent advances in the logical semantics of LCC to give an account of the restriction operator in Intuitionistic Linear Logic. This allows us to express as a Linear Logic theory, a notion of space similar to what has recently been introduced by Gabbay and Pitts in modal logics. It also permits to relate more closely the pi-calculus and CC paradigms that people have wanted to compare for a long time.