Abstract
We prove that, if Banach spaces X and Y are δ-average rough, then their direct sum with respect to an absolute norm N is δ/N(1,1)-average rough. In particular, for octahedral X and Y and for p in (1,∞), the space X⊕pY is 21−1/p-average rough, which is in general optimal. Another consequence is that for any δ in (1,2] there is a Banach space which is exactly δ-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and ∞-sums.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
