Chapter 1. Introduction Chapter 2. Definitions and Preliminary Results Chapter 3. Repeated Limit Conditions Chapter 4. Examples Chapter 5. Introverted Subspaces Chapter 6. Banach Algebras of Operators Chapter 7. Beurling Algebras Chapter 8. The Second Dual of 1 (G, ω) Chapter 9. Algebras on Discrete, Abelian Groups Chapter 10. Beurling Algebras on F 2 Chapter 11. Topological Centres of Duals of Introverted Subspaces Chapter 12. The Second Dual of L 1 (G, ω) Chapter 13. Derivations into Second Duals Chapter 14.