On universally left-stability of -isometry

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Let ... be two real Banach spaces and ... . A map ... is said to be a standard -isometry if ... for all ... and with ... . We say that a pair of Banach spaces ... is stable if there exists ... such that, for every such and every standard -isometry... , there is a bounded linear operator ... so that ... for all ... is said to be universally left-stable if ... is always stable for every Y (X). In this paper, we show that if a dual Banach space ... is universally left-stable, then it is isometric to a complemented ... -closed subspace of ... for some set [Gamma], hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant. (ProQuest: ... denotes formulae and non-USASCII text omitted)