1. The problem statement, all variables and given/known data Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. 2. Relevant equations 3. The attempt at a solution Assuming S is finite means that S is a closed set (a finite subset of a metric space is closed). I think that this will help to prove span(S) is a closed set but I am a bit stuck.