1. An extremely useful abstract closure principle. Suppose X is a vector space over R and Summary: 1. An extremely useful abstract closure principle. Suppose X is a vector space over R and | · | : X [0, ] is such that (i) |cx| = |c||x| whenever c R and x X; (ii) |x + y| |x| + |y| whenever x, y X. (If |x| < for each x X we say | · | is a seminorm on X; obviously, a norm on X is a seminorm on X.) For each a X and 0 < r < let U(a, r) = {x X : |x - a| < r} and let B(a, r) = {x X : |x - a| r}. As should come as no surprise, one calls U(a, r) the open ball with center a and radius r and one calls B(a, r) the closed ball with center a and radius r. We declare a subset U of X to be open if for each a U there is r (0, ) such that U(a, r) U. It is a simple matter which we leave to the reader to verify that the open sets are a topology on X which respect to which the open balls are open and the closed balls are closed. One easily verifies that this topology is Hausdorff if and only if |x| = 0 x = 0 whenever x X. Proposition 1.1. Suppose Y is a normed vector space, K : X Y and K is linear. Then K is continuous linear if and only if there is M [0, ) such that Collections: Mathematics