The Riesz Theory of compact operators comprises the study of eigenvalues, eigensubspaces, and generalized eigensubspaces of compact operators, culminating in a “Jordan decomposition” of sorts. Although the theory is well known in the context of Banach spaces, not many people know that it goes through undiminished in general linear topological spaces. Elucidating this aspect is the main goal in the present chapter. Its material is divided into three sections. First, §2.1 addresses elementary Riesz theory. It begins with §2.1.1, where we introduce and examine the class of compact linear operators on linear topological spaces. In §2.1.2, the focus shifts to eigenvalues, followed by §2.1.3, which provides a closer analysis of bounded operators. This sequence culminates in §2.1.4, where the preceding ideas converge in the study of compact perturbations of the identity. Second, §2.2 offers an algebraic interlude focused on range and null-space sequences associated with linear operators. Here, topological considerations are set aside to highlight the purely algebraic aspects of the theory. Third, in §2.3, we advance to higher-level Riesz theory. The central result demonstrates that, on linear topological spaces, compact perturbations of the identity exhibit behavior analogous to that of operators on finite-dimensional vector spaces: specifically, their range and null-space sequences eventually stabilize. After establishing this striking fact, we explore its implications for the spectral properties of compact operators. As a concluding highlight, we prove that for 0 < p < 1, the only compact operator supported by the p-Banach space Lp([0, 1]) is the zero operator – a property even stronger than the space’s triviality of its dual.