Property $(H)$ in Lebesgue-Bochner function spaces

The near Radon-nikodym Property in Lebesgue-bochner Function Spaces

Let X be a Banach space and (Ω,Σ, λ) be a finite measure space, 1 ≤ p < ∞. It is shown that L(λ,X) has the Near Radon-Nikodym property if and only if X has it. Similarly if E is a Köthe function space that does not contain a copy of c0, then E(X) has the Near Radon-Nikodym property if and only if X does.

Strong Barrelledness Properties in Lebesgue-Bochner Spaces

If (Ω,Σ, μ) is a finite atomless measure space and X is a normed space, we prove that the space Lp(μ,X), 1 ≤ p ≤ ∞ is a barrelled space of class א0, regardless of the barrelledness of X. That enables us to obtain a localization theorem of certain mappings defined in Lp(μ,X). By “space” we mean a “real or complex Hausdorff locally convex space”. Given a dual pair (E,F ), as usual σ(E,F ) denotes...

Extreme points of the unit cell Lebesgue-Bochner function spaces

Nonsquareness in Musielak-Orlicz-Bochner Function Spaces

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk ,...

COMPLEMENTED COPIES OF l AND PELCZYNSKI’S PROPERTY (V) IN BOCHNER FUNCTION SPACES

Let X be a Banach space and (fn)n be a bounded sequence in L (X). We prove a complemented version of the celebrated Talagrand’s dichotomy i.e we show that if (en)n denotes the unit vector basis of c0, there exists a sequence gn ∈ conv(fn, fn+1, . . . ) such that for almost every ω, either the sequence (gn(ω)⊗en) is weakly Cauchy in X⊗̂πc0 or it is equivalent to the unit vector basis of l. We the...