Measurability and Pettis integration in Hilbert spaces.
In the first section of this paper there are given criteria for strict convexity and smoothness of the Bochner-Orlicz space with the Orlicz norm as well as the Luxemburg norm. In the second one that geometrical properties are applied to the characterization of metric projections and zero mean valued best approximants to Bochner-Orlicz spaces.
Let E be a Banach space. Let be the Sobolev space of E-valued functions on with the norm . It is proved that if then there exists a sequence such that ; ; and for m = 1, 2,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding . In particular, the embedding into Besov spaces is proved, where for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada....
We show that the numerical index of a -, -, or -sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.