Let be normed linear space, a subspace of . The closure of , , is a linear subspace of .
We use the “sequential” equivalent definition of closure, rather than the one using open balls: is the set of all limits of all convergent sequences of points in . Let , . There is a sequence in such that . Similarly there is a sequence in which converges to .
Then is a sequence in that converges to . is a sequence in that converges to .