A Remark on the Stability of Approximative Compactness - Zhenghua Luo; Longfa Sun; Wen Zhang (Journal of Function Spaces)
We study the stability of approximative τ-compactness, where τ is the norm or the weak topology. Let Λ be an index set and for every λ∈Λ, let Yλ be a subspace of a Banach space Xλ. For 1≤p<∞, let X=⊕lpXλ and Y=⊕lpYλ. We prove that Y (resp., BY) is approximatively τ-compact in X if and only if, for every λ∈Λ, Yλ (resp., BYλ) is approximatively τ-compact in Xλ.