SPACES OF COMPACT OPERATORS AND THEIR DUAL SPACES 205 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo LIII (2004), pp. 205-224 SPACES OF COMPACT OPERATORS AND THEIR DUAL SPACES SHEM AYWA ∗ – JAN H. FOURIE ∗∗ The w ′ -topology on the space L ( X , Y ) of bounded linear operators from the Banach space X into the Banach space Y is discussed in [10]. Let L w ′ ( X , Y ) denote the space of all T ∈ L ( X , Y ) for which there exists a sequence of compact linear operators (T n ) ⊂ K ( X , Y ) such that T = w ′ − lim n T n and let ||| T ||| := inf {sup n ‖T n ‖ : T n ∈ K ( X , Y ), T n w ′ → T }. We show that (L w ′ , |||·|||) is a Banach ideal of operators and that the continuous dual space K ( X , Y ) ∗ is complemented in (L w ′ ( X , Y ), ||| · |||) ∗ . This results in necessary and sufficient conditions for K ( X , Y ) to be reflexive, whereby the spaces X and Y need not satisfy the approximation property. Similar results follow when X and Y are locally convex spaces. 1. Introduction and Notation. The question of the complementation of the dual space K ( X , Y ) ∗ of the space of compact operators K ( X , Y ) in the dual space L ( X , Y ) ∗ of L ( X , Y ) (the space of bounded linear operators from X into Y ) when X and Y are Banach spaces, was more or less settled by a result of J. Johnson (in 1979). Johnson proved in [9] that if Y is a Banach space having the bounded approximation property then the annihilator K ( X , Y ) ⊥ in the (continuous) dual space L ( X , Y ) ∗ is the kernel of a projection on L ( X , Y ) ∗ . The range space of the projection is isomorphic to the dual space K ( X , Y ) ∗ . Although many examples of Banach spaces which fail the approximation property became known after Enflo’s example in 1973, the fact that the classical spaces have ∗ Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. ∗∗ Financial support from the NRF and Potchefstroom University is greatly acknowledged. 2000 Mathematics Subject Classification: 47B10; 46B10; 46A25