Uniformly non-ln(1), locally uniformly non-ln(1) and non-ln(1) properties in Orlicz–Bochner function spaces endowed with the Orlicz norm

On non-midpoint locally uniformly rotund normability in Banach spaces

We will show that if X is a tree-complete subspace of ∞ , which contains c 0 , then it does not admit any weakly midpoint locally uniformly convex renorming. It follows that such a space has no equivalent Kadec renorming. 1. Introduction. It is known that ∞ has an equivalent strictly convex renorming [2]; however, by a result due to Lindenstrauss, it cannot be equivalently renormed in locally u...

Nonsquareness and Locally Uniform Nonsquareness in Orlicz-Bochner Function Spaces Endowed with Luxemburg Norm

On the uniformly normal structure of Orlicz spaces with Orlicz norm*

We prove that in Orlicz spaces endowed with Orlicz norm the uniformly normal structure is equivalent to the reflexivity.

Extreme Points and Rotundity in Musielak-Orlicz-Bochner Function Spaces Endowed with Orlicz Norm

and Applied Analysis 3 Put LM X { u ∈ XT : ρM λu < ∞ for some λ > 0 } . 1.5 Then the Musielak-Orlicz-Bochner function space ‖u‖ inf k>0 1 k [ 1 ρM ku ] 1.6 is Banach space. If X R, LM R is said to be Musielak-Orlicz function space. Set K u { k > 0 : 1 k ( 1 ρM ku ) ‖u‖ } . 1.7 In particular, the set K u can be nonempty. To show that, we give a proposition. Proposition 1.1. If limu→∞ M t, u /u ∞...

Almost Euclidean subspaces of lN1 via expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ RN of dimension (1− o(1))N such that for every x ∈ X, (logN)−O(log log logN) √ N ‖x‖2 6 ‖x‖1 6 √ N ‖x‖2. If we are allowed to use N1/ log logN 6 No(1) random bits and dim(X) > (1 − η)N for any fixed constant η, the lower bound can be further improved to (logN)−O(1) √ N‖x‖2. Through known connections...