We introduce a vector-valued version of uniform algebra, called the function space over algebra. The diameter two properties algebra on an infinite compact Hausdorff are investigated. Every nonempty relatively weakly open subset unit ball $$A(K, (X, \tau ))$$ dimensional has two, where $$\tau $$ is locally convex topology Banach X compatible to dual pair. Under assumption equipped with norm bei...

It is known that there are several approaches to define a Sobolev class for Banach valued functions. We compare the usual definition via weak derivatives with Reshetnyak–Sobolev space and Newtonian space; in particular, we provide sufficient conditions when all three agree. Also, revise difference quotient criterion property of Lipschitz mapping preserve it acting as superposition operator.

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L X , where X is a Banach space and 1 ≤ p < ∞, and extend the result to vector-valued Banach function spaces EX , where E is a Banach function space with order continuous norm. Let X be a Banach space. The problem of describing the compact sets in the Lebesgue-Bochner spaces LpX , ...