Is it possible to have probabilistic metric space (S,F,T) be a topological vector space too? In specific way, the probabilistic metric space is Menger and does not have a norm, however with Menger space we know that a probabilistic metric exists which may redefined the properties of TVS (absolutely convex, absorbing, ... ) with the probabilistic metric induced by Menger space.
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3$\begingroup$ This question is very poorly formulated. Obviously a set of continuum cardinality can bear a TVS structure in an infinite variety of ways, and can bear a probabilistic metric space structure in an infinite variety of ways. Do you want particular relations to hold between the two types of structure? $\endgroup$– Todd TrimbleCommented Oct 28, 2017 at 20:36
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$\begingroup$ I see some votes to close the question. I think that those votes were cast a little bit too quickly. I would recommend that we wait at least a couple of hours to let the OP reformulate their question, as suggested by Todd Trimble. $\endgroup$– André HenriquesCommented Oct 28, 2017 at 20:48
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$\begingroup$ the question makes sense if TVS does not admit a norm . $\endgroup$– behrad mahboobiCommented Oct 29, 2017 at 9:19
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$\begingroup$ @AndréHenriques ... "wait a couple of hours" ... done ... but putting it "on hold" will keep the problem here until the OP comes back to explain. Since it is the weekend, it may take 2 or 3 days. Maybe he should include a reference like en.wikipedia.org/wiki/Probabilistic_metric_space to explain what is a probabilistic metric space. $\endgroup$– Gerald EdgarCommented Oct 29, 2017 at 13:00
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$\begingroup$ Probabilistic metric space is quite different by probability space. Specially menger space has widely investigated due to it apllication in fixed point theory. So i guess you assume you want to make or redefine TVS by induced t-norm? $\endgroup$– behrad mahboobiCommented Oct 29, 2017 at 19:18
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