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Consider a locally convex topological vector space. Now, using Minkowski functional (which are semi-norms) we can define a family of semi-norms indexed by the local basis (convex,balanced, absorbing) of the locally convex topological vector space. Now they can be used to define a topology and a locally convex topological vector space. How do I check that the new topology is same as the old one ?

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They do define the same topology.

This is Theorem 1.37 (page 27) in Rudin's Functional Analysis.

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