Proximinality of subspaces and the quotient lifting property

Abstract

The Quotient Lifting Property (QLP) for pairs of spaces and its relation to the proximinality of subspaces are studied. Even though this property is not hereditary, for finite-dimensional subspaces of $L^1(\mu )$, we show that for a choice of a basis, all subspaces spanned by any sub-collection from the base, generate subspaces with QLP. Spaces of dimension greater than two with the total quotient lifting property are shown to be Hilbert spaces.

Keywords:

• Quotient lifting property
• proximinality
• complemented subspaces

2000 Mathematics Subject Classifications:

• Primary 47L05
• 46B28
• 46B25

Acknowledgments

The authors are grateful to the referee for a careful reading and many useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).