 
Summary: 1. An extremely useful abstract closure principle.
Suppose X is a vector space over R and
 ·  : X [0, ]
is such that
(i) cx = cx 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
