The space 𝐿^{𝜔} and convex topological rings

Introduction. The motive for investigating the class L w of functions belonging to all Z>-classes has no measure-theoretic origin: it was our desire to discover whether or not in every convex metric ring 1 R one could find a system { U] of convex neighborhoods of 0 having the property that/, g E U implies f g G U. We show here that Z, w has no proper convex open set U containing 0 and satisfying the relation UUQ U, thus supplying the desired counter-example.

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