English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In a previous paper of the first two authors [GM] the space JT. was constructed as a James space over a tree with infinitely many branching points. It was proved that the predual B. of JT,, fails the "point of continuity property." In the present paper we show that B. has the so-called "convex point of continuity property" thus answering a question of Edgar and Wheeler [EW] in the negative. 1. Definitions, notations and preliminaries. Recall [BR] that a Banach space X has the point of continuity property (PCP) if for every weakly closed bounded subset C of X there is x E C such that the weak and norm topologyrestricted to C-coincide at x. An equivalent formulation goes as follows (compare [B, Proposition 1]): For every bounded subset C of X and for E > 0 there is a relatively weakly open subset U of C such that the norm-diameter of U is less than Previously, J. Bourgain [B] introduced (under the name of property (*)) the following weaker concept: A Banach space X has the convex point of continuity property (CPCP) if every closed convex bounded subset C of X has a point x where the relative weak and norm topologies coincide. Again this may be rephrased as follows: For every convex bounded subset C of X and E > 0 there is a relatively weakly open subset U of C of norm-diameter less than E. The question whether PCP and CPCP are in fact equivalent remained open and was explicitly asked in [EW]. We shall show that the space Boo furnishes an example with CPCP but failing PCP. Recall from [GM] the definition of JToo: We consider the tree with infinitely many branching points 00

A counterexample to a problem on points of continuity in Banach spaces