Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed.

Is there some description of class of convex sequential topologies?

There is a description of C-sequential topologies (i.e. every convex sequentially open subset is open; for example, http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002307847&physid=PHYS_0280), but I can't find anything about convex sequential topologies.

In particular,

1) Is there a convex sequential, but not sequential topology?

2) Let $X$ bу a topological vector space such that all linear functionals on $X$ are continious. Can the topology on $X$ be convex sequential, but not sequential?

EDIT: I found an answer to questions 1 and 2. Let $X$ be an infinite-countable dinensional vector space, $X^\#$ its algebraic dual space. Then the weak topology relative to the duality $X \leftrightarrow X^\#$ is convex sequential, but not sequential. However, the main question is not solved.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.