On operators which attain their norm
The following problem is considered. LetX andY be Banach spaces. Are those operators fromX toY which attain their norm on the unit cell ofX, norm dense in the space of all operators fromX toY? It is proved that this is always the case ifX is reflexive. In general the answer is negative and it depends on some convexity and smoothness properties of the unit cells inX andY. As an application a refinement of the Krein-Milman theorem and Mazur’s theorem concerning the density of smooth points, in the case of weakly compact sets in a separable space, is obtained.
KeywordsBanach Space Extreme Point Convex Space Normed Linear Space Separable Banach Space
Unable to display preview. Download preview PDF.
- 7.Kadec, M. I., 1963, Some problems in the geometry of Banach spaces, Dissertation, Moscow University.Google Scholar