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- Suppose that is a vector space and let be a non-empty collection of convex, balanced, and absorbing subsets of
Then the set of all positive scalar multiples of finite intersections of sets in forms a neighborhood base at the origin for a locally convex TVS topology on (en)
- Let be a Fréchet space over the field
Then the following are equivalent:
# does admit a continuous norm .
# contains a vector subspace that is TVS-isomorphic to
# contains a complemented vector subspace that is TVS-isomorphic to (en)
- Suppose that is a vector space and let be a filter base of subsets of such that:
# Every is convex, balanced, and absorbing;
# For every there exists some real satisfying such that
Then is a neighborhood base at 0 for a locally convex TVS topology on (en)
- Every complete metrizable TVS with the Hahn-Banach extension property is locally convex. (en)
- Let be a linear operator between TVSs where is locally convex . Then is continuous if and only if for every continuous seminorm on , there exists a continuous seminorm on such that (en)
- If is a TVS and if is a linear functional on , then is continuous if and only if there exists a continuous seminorm on such that (en)
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