An element "lives" in a space

I have seen the expression, "X is the Hilbert space in which the element x lives". As a native speaker, this seems quite sloppy to me. Is there a more succinct way to formulate this expression?


Solution 1:

You could say:

X is in the Hilbert space to which the element x belongs.

Though I don't think using "lives" is necessarily wrong (it is a bit informal, though).