Linear spaces, absolute retracts, and the compact extension property

Abstract

We formulate a "partial realization" property and prove that this property is equivalent to the compact extension property. In addition, we prove that a linear space L L has the compact extension property if and only if L L is admissible if and only if L L has the σ \sigma -compact extension property. This implies that for a σ \sigma -compact linear space L L , the following statements are equivalent: (1) L L is an absolute retract, (2) L L has the compact extension property, and (3) L L is admissible. Finally, we prove that if there exists a linear space which is not an absolute retract then there is an admissible linear space which is not an absolute retract.