Pointwise sequential compactness and weak compactness in spaces of contents
Pointwise relative sequential compactness and sequential precompactness (sometimes called ''conditional compactness'') of sets of separable-valued contents are characterized. Using a theorem of Simon of the \(type\) ''\(<x',x_ n>\to <x',x>\) for certain x' implies weak-lim \(x_ n=x''\) a connection between pointwise convergence (in the weak topology of the range space) and weak convergence for contents with totally bounded ranges is obtained. Both together yield a criterion for weak relative sequential compactness (sequential precompactness) and, using the Eberlein-Šmulian theorem, a criterion for weak relative (countable) compactness of sets of contents, which have separable and totally bounded ranges. The assumption on the (locally convex) range space E depends only on the dual pairing \(<E,E'>\); e.g. E can be endowed with the weak as well as the strong topology of a Fréchet space.
Pointwise relative sequential compactness
weak convergence for contents with totally bounded ranges
scientific article; zbMATH DE number 3929713
scientific article