COMPATIBLE LOCALLY CONVEX TOPOLOGIES ON NORMED SPACES: CARDINALITY ASPECTS

For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to $\mathfrak{c}$.

THE STRONG DUAL OF MEASURE ALGEBRAS WITH CERTAIN LOCALLY CONVEX TOPOLOGIES

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.