[math-fun] Hilbert space puzzle

Let H denote the Hilbert space of all square-summable sequences of real numbers. (With the usual inner product <x,y> = Sum_{k=1...oo} x_k y_k, and corresponding norm ||x|| = <x,x>^(½) and metric d(x,y) = ||x-y||.) H is a vector space endowed with the topology determined by its metric. Now define L := {x in H | all but finitely many coordinates x_k are 0, and Sum_{k=1...oo} x_k = 0}. L is clearly a vector subspace of H. PUZZLE: Describe explicitly the closure Cl(L) of L. --Dan