On Thursday 06 September 2012 19:05:53 Dan Asimov wrote:
Let H denote the Hilbert space of all square-summable sequences of real numbers. ... L := {x in H | all but finitely many coordinates x_k are 0, and Sum_{k=1...oo} x_k = 0}. ... PUZZLE: Describe explicitly the closure Cl(L) of L.
... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... ... spoiler space ... It's all of H. Let x be any element of h and for each k define x[k] to agree with x for its first k coordinates, and have the next k coordinates equal, their value chosen to make the sum of the coordinates 0. The remaining (infinity-2k) coordinates are zero. Obviously x[k] is in L. Writing y\k for the result of zeroing the first k coordinates of y, ||x-x[k]|| = ||x\k-x[k]\k|| <= ||x\k|| + ||x[k]\k|| where the first summand obviously -> 0 as k->oo because otherwise x isn't square-summable and the second also -> 0 because it's at most ||x||/sqrt(k). -- g