Publisher Summary This chapter discusses the theory of linear compact operators, which is the direct generalization of Fredholm's results. The theory is extremely important in applications—as indeed is the allied theory of nonlinear compact operators. It is easy to see that every compact operator is bounded. In finite dimensions, every bounded operator is compact, but this is not true in infinite dimensions. The chapter also considers some operators that commonly occur in applications and enquire whether they are compact. The examples illustrate two standard methods of proving compactness. Each of the compact operators may be uniformly approximated by operators of finite rank—that is, with finite dimensional range, and it is tempting to speculate that this may be true for an arbitrary compact operator on a Banach space—a compact operator is the sum of an operator of finite rank and an operator of arbitrarily small norm. The Fredholm Alternative is discussed. The proof of the corresponding result for integral equations provides a powerful technique for establishing the existence of solutions to boundary value problems for elliptic partial differential equations. Compact Self-adjoint Operators are discussed. The chapter concludes with a discussion of the numerical solution of linear integral equations.