Abstract
We prove that every probabilistic normed space, either according to the original definition given by erstnev, or according to the recent one introduced by Alsina, Schweizer and Sklar, has a completion.
Highlights
A real or complex nornred linear space admits a completion, namely, given a normed linear space (y, Il . ll ), there exists another linear space (y', ll ll ') such that V' is isometric to a ddnse subspace of I/. It was proved by Mu5tari [2], Sherwooci ([7], [8j) and Sempi [5] thai a probabilistic metric space has a completion
The proof will be given in both cases
In order to prove that V' is a lineal space, iet p' and q' be elements of V' and let {p"} and {g"} be Cauchy sequences of elements of V with {p*} e p' and iq"} € q'
Summary
A real or complex nornred linear space admits a completion, namely, given a normed linear space (y, Il . ll ), there exists another linear space (y', ll ll ') such that V' is isometric to a ddnse subspace of I/.It was proved by Mu5tari [2], Sherwooci ([7], [8j) and Sempi [5] thai a probabilistic metric space has a completion. A real or complex nornred linear space admits a completion, namely, given a normed linear space Ll ), there exists another linear space (y', ll ll ') such that V' is isometric to a ddnse subspace of I/.
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