A normability condition for Frechet spaces (Q1819316)

Let X be a Frechet space with a continuous norm p, and let \(U_ p=\{x:p(x)\leq 1\}.\) The authors prove that if every continuous linear functional on X which is bounded on \(U_ p\) attains its supremum on \(U_ p\), then p generates the original topology of X, which is therefore a reflexive Banach space. Now let V be a closed bounded absolutely convex subset of X, with associated seminorm p. A point \(x\in V=\{x:p(x)=1\}\) is said to be attainable if there exists a continuous linear functional f such that \(f(x)=1\) and f(y)\(\leq p(y)\) for all \(y\in V\). If V has non-empty interior, then by the Hahn-Banach Theorem every point of V is attainable. The authors prove a converse result: if V is in addition total, and every point of V is attainable, then V has non-empty interior, and hence X is a Banach space. Examples are given to show that metrizability cannot be dropped.