7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

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Weak vector space topologies Theorem Theorem 3.10. Let X be a vector space and X ′ a separating vector space of linear functional on X. Then the X ′ -topology makes X a locally convex topological vector space whose dual is X ′ . The proof is based on the following lemma of independent interest: Lemma Let l1,l2,... ,ln and l be linear functional on the vector space X. The following are equivalent: (a) There are scalars α1,α2,... ,αn such that l = α1l1 + α2l2 + · · · + αnln. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
Weak vector space topologies Theorem Theorem 3.10. Let X be a vector space and X ′ a separating vector space of linear functional on X. Then the X ′ -topology makes X a locally convex topological vector space whose dual is X ′ . The proof is based on the following lemma of independent interest: Lemma Let l1,l2,... ,ln and l be linear functional on the vector space X. The following are equivalent: (a) There are scalars α1,α2,... ,αn such that l = α1l1 + α2l2 + · · · + αnln. (b) There is a γ > 0 such that |l(x)| ≤ γ max1≤i≤n |li(x)| for all x ∈ X. Klaus Thomsen 7. Consequences of the Hahn-Banach theorems
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Page 31 and 32: Weak vector space topologies Lemma
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7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

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