Asplund Function Asplund Space Weakly LindelÖf Determined Space Projectional Resolution Of The Identity Locally Uniformly Rotund Norm
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 26, No 4, (2000), 287p-308p
We further develop the theory of the so called Asplund functions,
recently introduced and studied by W. K. Tang. Let f be an Asplund
function on a Banach space X. We prove that (i) the subspace
Y := sp ∂f(X) has a projectional resolution of the identity, and that (ii) if
X is weakly Lindel¨of determined, then X admits a projectional resolution of
the identity such that the adjoint projections restricted to Y form a projectional
resolution of the identity on Y , and the dual X* admits an equivalent
dual norm such that its restriction to Y is locally uniformly rotund.
*Supported by the Grants AV ˇCR 101-97-02, 101-90-03, GA ˇCR 201-98-1449, and by the
Grant of the Faculty of Civil Engineering of the Czech Technical University No. 2003.