Quadruplex Real-Time PCR Assay for Detection and Identification of Vibrio cholerae O1 and O139 Strains and Determination of Their Toxigenic Potential
- 生命科学－已发表论文 
By a ball-covering B of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering, B is called minimal if its cardinality B-# is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n is an element of N with k <= n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.