Introduction

The classical Fredholm theory is due to Erik Ivar Fredholm (1866–1927), after whom it is named. The first recorded reference to the theory appears to be in his letter (1899) to Mittag-Leffler, dated 8 August. A short account of his theory was given in (Fredholm, 1980; cf. also 1902a, b). The definitive account appeared in (Fredholm, 1903; cf. Fredholm, 1930, pp. viii–ix; Bôcher, 1914, p. 25; Whittaker & Watson, 1927, p. 213; Hellinger & Toeplitz, 1927, p. 1354). Other accounts of the theory are given, for instance, in (Bôcher, 1914, pp. 29–46; Goursat, 1923, pp. 368–438; 1964, pp. 46–116; Lovitt, 1924, pp. 23–72; Courant & Hilbert, 1924, pp. 99–141; 1931, pp. 96–138; 1953, pp. 112–63; Hellinger & Toeplitz, 1927, pp. 1370–6; Kowalewski, 1930, pp. 91–247; Smithies, 1958, pp. 65–78; Zabreyko et al, 1975, pp. 26–56). Most of these accounts deal only with real functions and real Fredholm-kernels, but the transition to complex functions and complex Fredholm-kernels is routine.

The word ‘function’ is currently used with various shades of meaning. We shall follow Moss & Roberts (1968, p. 13). In this connection, if φ is a function from a set S to a set T, then I call S the source, and T the destination, of φ.