Compact maps constitute a class of maps to which we can extend many of the results which are valid for maps between finite-dimensional spaces. Compactness plays a central role in the infinite-dimensional extension of degree theory (Leray–Schauder degree, see Chap. 3) and in fixed point theory (see Chap. 4). The needs of problems in the calculus of variations and in nonlinear functional equations led to the class of operators of monotone type, which provides a broader framework than the class of compact maps.