Abstract
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.
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© 1984 Springer-Verlag New York Inc.
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Luecking, D.H., Rubel, L.A. (1984). Duality of H(G)—The Case of the Unit Disc. In: Complex Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8295-9_6
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DOI: https://doi.org/10.1007/978-1-4613-8295-9_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90993-6
Online ISBN: 978-1-4613-8295-9
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