Ball generated property of direct sums of Banach spaces
Jan-David Hardtke

TL;DR
This paper investigates the stability of the ball generated property (BGP) in Banach spaces under certain direct sum constructions, extending previous results to a broader class of norms.
Contribution
It proves that the BGP is preserved under direct sums of Banach spaces with respect to a wide class of smooth, normalized norms on ^2, generalizing earlier stability results.
Findings
BGP is stable under direct sums with specific smooth norms.
Extension of BGP stability beyond _0- and a_p-sums.
Applicable to a broad class of absolute, normalized, smooth norms.
Abstract
A Banach space is said to have the ball generated property (BGP) if every closed, bounded, convex subset of can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) - and -sums for . We will show here that for any absolute, normalised norm on satisfying a certain smoothness condition the direct sum of two Banach spaces and with respect to enjoys the BGP whenever and have the BGP.
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