Abstract
With this metric, F is a complete linear metric space and convergence of a sequence in F is equivalent to uniform convergence of the sequence of integral functions in any circle of finite radius [1, Theorems 1 and 3 ]. Being a locally convex space in the concerned metric, the Hahn-Banach extension theorem for continuous linear functionals remain valid (for a direct proof see [2, Theorems 3 and 4]). The topology of r can also be specified by the family of norms
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