Department of Mathematics2024-11-0920020010-135410.4064/cm91-2-22-s2.0-77953697627http://dx.doi.org/10.4064/cm91-2-2https://hdl.handle.net/20.500.14288/11369A Banach space X has property (E) if every operator from X into c0 extends to an operator from X** into c0; X has property (L) if whenever K ⊆ X is limited in X**, then K is limited in X; X has property (G) if whenever K ⊆ X is Grothendieck in X**, then K is Grothendieck in X. In all of these, we consider X as canonically embedded in X**. We study these properties in connection with other geometric properties, such as the Phillips properties, the Gelfand-Phillips and weak Gelfand-Phillips properties, and the property of being a Grothendieck space. © 2002, Instytut Matematyczny. All rights reserved.MathematicsJournal Articlehttps://www.scopus.com/inward/record.uri?eid=2-s2.0-77953697627&doi=10.4064%2fcm91-2-2&partnerID=40&md5=5a11059129121bea3ecfc04ea682ad9aQ311967