Norm topology

Abstract

If the distance between two operators A and B is defined to be ‖ A - B ‖, the set of all operators on a Hilbert space becomes a metric space. Some of the standard metric and topological questions about that space have more interesting answers than others. Thus, for instance, it is no more than minimum courtesy to ask whether or not the space is complete. The answer is yes. The proof is the kind of routine analysis every mathematician has to work through at least once in his life; it offers no surprises. The result, incidentally, has been tacitly used already. In Solution 72, the convergence of the series $\sum\nolimits_{{n = 0}}^{\infty } {{A^{n}}}$ was inferred from the assumption ‖ A ‖ < 1. The alert reader should have noted that the justification of this inference is in the completeness result just mentioned. (It takes less alertness to notice that the very concept of convergence refers to some topology.)