Anderson's lemma

Appendix ETopological vector spaces1. Locally convex topological vector spacesA vector space X equipped with a topology is said to be a topological vectorspace (TVS) if(i) (x 1 , x 2 ) ↦→ x 1 + x + 2 is continuous as a map from X ⊗ X into X(ii) (λ, x) ↦→ λx is continuous as a map from R ⊗ X into XA TVS is said to be locally convex if it is Hausdorff and if every neighborhoodof 0 contains a convex neighborhood of 0. Abbreviate “locally convextopological vector space” to lcTVS. Example. A TVS whose topology is defined by a norm is a lcTVS. The□ balls centered at the origin are convex. Example. Let X be a vector space and Ɣ be a collection of linear functionsfrom X into R that separate points (that is, if x and y are distinct points of Xthen f (x) ≠ f (y) for at least one f in Ɣ). Then the weakest topology on Xmaking all functions in Ɣ continuous makes X a lcTVS. The sets{x ∈ X : | f i (x)| sup λxx∈FIn particular, X ∗ separates points of X.If f is a convex, lower semi-continuous map from X into R ∪ {∞}, theepigraphepi( f ) ={(x, t) ∈ X ⊗ R : t ≥ f (x)}is a closed, convex subset of the lcTVS X ⊗ R. The hyperplanes that separateepi( f ) from points (x, s) with s < f (x) define a collection of continuous linearfunctionals on X whose pointwise supremum equals f . That is, f (x) = sup{x ∗ (x) : f ≥ x ∗ on X}Asymptopia: 4Jan99 c○David Pollard 1