On strong proximinality in normed linear spaces

The paper deals with strong proximinality in normed linear spaces. It is proved that in a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

A Remark on the Stability of Approximative Compactness

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.