A Remark on the Stability of Approximative Compactness

Abstract EN

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 Y λ be a subspace of a Banach space X λ .

For 1 ≤ p < ∞ , let X = ⊕ l p X λ and Y = ⊕ l p Y λ .

We prove that Y (resp., B Y ) is approximatively τ -compact in X if and only if, for every λ ∈ Λ , Y λ (resp., B Y λ ) is approximatively τ -compact in X λ .