An algebraic version of the multiplication property of the Fredholm index (Q1336409)

Let \(X\), \(Y\), \(Z\) be vector spaces and \(T: X\to Y\), \(S: Y\to Z\) everywhere defined linear operators. It is shown that \[ \dim\ker ST+ \text{codim ran } S+ \text{codim ran } T= \text{codim ran } ST+ \dim\ker S+ \dim\ker T. \] The well-known index formula \(\text{ind } ST= \text{ind } T+\text{ind } S\) is thus seen to hold not only when the quantities involved are all finite, but also whenever the cardinal arithmetic makes sense (for example, when \(S\) and \(T\) both have finite-dimensional kernels).