Fredholm operator cokernel and index

Let $X$ and $Y$ be Banach spaces over the same scalar field $\mathbb{F}$ (Banach space) and let $T:X\to Y$ be a bounded linear operator (A bounded linear operator between normed spaces). Then $T$ is a Fredholm operator when all three of the following hold:

1. $\ker T$ is finite dimensional, that is, admits an ordered basis of finite length (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis);
2. $\mathrm{ran}T$ is a closed subspace of $Y$ (Linear subspace of a vector space);
3. the cokernel $\mathrm{coker}T:=Y\mathrm{/}\mathrm{ran}T$ is finite dimensional (The quotient vector space (X/M), its cosets, and the quotient map (q:X\to X/M), Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis).

Here the cokernel is the algebraic quotient vector space; it also carries the quotient seminorm of The quotient seminorm (|x+M|{X/M}=\inf{m\in M}|x+m|=\operatorname{dist}(x,M)), which is a norm when the range is closed (The quotient seminorm is a norm exactly when the subspace is closed), but the dimension in clause 3 is the vector-space dimension of the quotient and does not depend on that norm.

For a Fredholm operator $T$ the index of $T$ is the integer

$\mathrm{ind}T:={\dim }_{\mathbb{F}}\ker T-{\dim }_{\mathbb{F}}\mathrm{coker}T$

Here subtraction is in $\mathbb{Z}$, not in $\mathbb{N}$: write $a={\dim }_{\mathbb{F}}\ker T$ and $b={\dim }_{\mathbb{F}}\mathrm{coker}T$ and identify a natural $n$ with $[(n,0)]$. Precisely,

$\mathrm{ind}T=[(a,b)]=[(a,0)]+(-[(b,0)])\in \mathbb{Z}$

by The integers as equivalence classes of pairs of naturals and Arithmetic on the integers. Both dimensions are natural numbers, so this class is defined even when $a<b$; the index may be positive, negative or zero.

Two remarks on the definition. The closedness of the range is listed explicitly as a hypothesis of the definition rather than extracted from the other clauses. And no property of the index is asserted here; additivity, local constancy and invariance under compact perturbations are separate theorems.