The topological structure of a certain Menger space |
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Authors: | F Bahrami M Nikfar |
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Institution: | Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156, Iran |
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Abstract: | In this paper we will reconsider the topological structure of Menger probabilistic normed spaces (briefly PN-spaces) under the t-norm M. We will prove that this topology is compatible with the topology induced by a countable and separating family of semi-norms, and hence the well-known theorems of classical functional analysis (such as the principle of uniform boundedness, open mapping and closed graph theorems) are valid in this context also. We will meanwhile obtain a method by which one may construct easily a large class of PN-spaces. Finally, using this method, we see that a certain subspace of bounded linear operators between PN-spaces, i.e. the class of strongly bounded linear operators, has a natural PN structure. |
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Keywords: | Probabilistic normed space Locally convex space Strongly bounded linear operator |
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