Duality of H(G)—The Case of the Unit Disc

We begin with a general result about linear functionals on a locally convex topological vector space. Let E have the topology generated by a family P of seminorms. For each non-empty finite set A = {‖•‖₁, ‖•‖₂,…, ‖•‖_n} ⊂ P, define$$ {\left\| x \right\|_A} = \mathop{{\max }}\limits_{{l \leqslant j \leqslant n}} \,{\left\| x \right\|_j} $$, x ∈ E. Then ‖•‖_A is a seminorm. Let P̃ = P ∪ {‖•‖_A: A is a non empty finite subset of P}; then P and P̃ generate the same topology on E (Exercise 2). Consequently, we may assume P = P̃ in the following proposition.