Results of \(B\)-complete and \(B_ r\)-complete locally convex spaces (Q1820380)

Let \(E\) be a Banach space and \(E'\) its conjugate space. The following terminology is used: A functional defined on the subspace M of E' is called to be almost continuous if for each neighbourhood of zero \(V\subset E\) the mapping \(f|_{(V^ 0\cap M)}\), where \(v^ 0\) is the polar of \(V\) in \(E'\), is continuous in the topology \(\sigma (E',E)|_{(V^ 0\cap M)}\). The subspace \(L\subset E'\) is almost closed if for each neighbourhood \(V\) of zero in \(E\) the set \(V^ 0\cap M\) is closed in the weak topology \(\sigma(E',E)\). The space \(E\) is \(B\)- complete if each almost closed subspace of \(E'\) is closed in the weak topology and \(B_ r\)-complete if in the space \(E'\) with the weak topology does not exists a proper everywhere dense almost closed subspace. The following conditions for \(B\)-completeness and \(B_ r\)-completeness in term of almost continuous linear functionals are proved: - \(E\) is \(B\)-complete if and only if each almost continuous linear functional defined on the almost closed subspace \(M\subset E'\) is continuous in the topology \(\sigma (E',E)|_ M.\) - \(E\) is \(B\)-complete \((B_ r\)-complete) if and only if for each almost closed subspace (every dense almost closed subspace) \(M\) in \(E'\) with the weak topology the factorspace \(E/M^ 0\) (the space \(E\)) is complete in the natural topology \(\nu(E/M^ 0,M)\) (\(\nu(E,M)\)). - If \(E\) is a barrelled space, then each almost closed subspace \(M\subset E'\) is a bornological space in the Mackey topology \(\tau (M,E/M^ 0)\) if and only if \(E\) is \(B\)-complete and \(M\) is quasireflexive in the topology \(t_ M.\) - The Schwartz space \(E\) is complete if and only if the topology of inductive limit \(t_{E'}\) and the Mackey topology \(\tau(E',E)\) coincides on \(E'\). - The barrelled Schwartz space is \(B\)-complete if and only if each almost closed subspace \(M\subset E\) is bornological in the Mackey topology \(\tau(M,E/M^ 0)\).