In this chapter we explore the algebraic and topological properties of a remarkable class of operators, whose definition is inspired by the very foundations of Riesz theory. Our starting point is to clarify what constitutes a natural notion of a Fredholm operator on a Hausdorff linear topological space. This task is undertaken in §3.1, building on insights from our earlier work on Riesz theory. Central to this discussion is the Fundamental Fredholm Decomposition (Theorem 3.1.7), which asserts that a Fredholm operator is essentially a linear homeomorphism augmented by finite-dimensional behavior. This perspective permeates the subject and is particularly evident in Theorem 3.1.10, the analogue of the classical Atkinson theorem for topological vector spaces. In this sense, it may rightly be regarded as the Fundamental Theorem of Fredholm Theory in the setting of topological vector spaces. In §3.2 we introduce and study the Calkin algebra of continuous linear operators modulo compact operators on topological vector spaces. The Fredholm index is then introduced in §3.3, with the goal of demonstrating its striking algebraic and topological properties, which deepen our understanding of linear operators and their spectra. We begin with the case of index zero in §3.3.1, followed by the scenario of negative index in §3.3.2. The crucial additivity of the index for Fredholm operators on topological vector spaces is stated in §3.3.4. This result is highly consequential, and immediate applications – such as the invariance of the Fredholm index under compact perturbations – are explored in §3.3.5. The proof of the additivity theorem is completed in §3.3.6. The continuity of the Fredholm index is examined in §3.4. Here we restrict attention to the most commonly occurring class of linear topological spaces: the p-Banach spaces, with 0 < p ≤ 1. For such spaces, Banach-space techniques provide a “p-norm topology” for the space of continuous linear operators. Within this setting, we show that the Fredholm index, viewed as a mapping from the space of Fredholm operators (equipped with the operator norm topology) into the integers with the discrete topology, is continuous. In fact, we prove a slightly stronger but equivalent statement: Fredholm operators of a fixed index form an open subset of the space of continuous linear operators. Significantly, we also provide an example showing that continuity of the index may fail for Fredholm operators on Fréchet spaces. In §3.5 we apply Fredholm theory to the study of the spectrum of a linear operator. In §2, dealing with Riesz theory in topological vector spaces, we established that the spectrum of a compact operator on an infinitedimensional Hausdorff linear topological space consists of the origin together with a (possibly empty) set of finite-multiplicity eigenvalues which, if infinite, form a sequence converging to zero. This basic spectral configuration motivates the pursuit of further spectral variants here. Our efforts culminate in §3.6, where we develop a general Fredholm theory applicable to operators between two (possibly distinct) linear topological spaces. The treatment in this chapter has thus far focused on the “diagonal” case, where a linear operator acts from a space into itself. However, it is of considerable interest to consider the “off-diagonal” case, allowing operators to act between different spaces. Extending Fredholm theory to this broader setting presents new challenges: in particular, operators may no longer be composable with themselves, a key feature in earlier proofs. Our approach is to use the previously developed theory as a stepping stone, showing that it essentially self-extends from the diagonal to the off-diagonal case via a matrix formalism. The core results are established in §3.6.1 for linear topological spaces and in §3.6.2 for quasi-Banach spaces, while §3.6.3 gathers various auxiliary results related to the development of a semi-Fredholm theory on quasi-Banach spaces.