Typically, a basic text on functional analysis will only make the briefest of references to general topological vector spaces, before restricting attention to the locally convex case or to Banach spaces. Thus most analysts are aware of the existence of non-locally convex spaces such as Lp (0, 1) for 0 < p < 1 but know very little about them. The neglect of nonlocally convex spaces is easily understood. The basic theory of Banach spaces, which sits at the core of modern functional analysis, may be said to depend on two major principles – the Hahn-Banach theorem and the Closed Graph theorem (which may be taken to include weaker theorems such as the Uniform Boundedness Principle). Working in non-locally convex spaces, even when they are complete and metrizable, requires doing without the Hahn- Banach theorem. The role of the Hahn-Banach theorem may be said to be that of a universal simplifier – infinite-dimensional arguments can be reduced to the scalar case by the use of the ubiquitous linear functional. Thus the problem with non-locally convex spaces is that of “getting off the ground.” This difficulty in even making the simplest initial steps has led some to regard non-locally convex spaces as simply uninteresting. It's our contention, which we hope to justify in these notes, that this attitude is mistaken and that with the aid of fresh techniques one can develop a rich and fulfilling theory.