A Remark on the Stability of Approximative Compactness

Abstract

We study the stability of approximativeτ-compactness, whereτis the norm or the weak topology. LetΛbe an index set and for everyλ∈Λ,letYλbe a subspace of a Banach spaceXλ. For1≤p<∞, letX=⊕lpXλandY=⊕lpYλ. We prove thatY(resp.,BY) is approximativelyτ-compact inXif and only if, for everyλ∈Λ,Yλ(resp.,BYλ) is approximativelyτ-compact inXλ.