Atiyah's K-Theory (Harvard lectures, 1964) ends with an appendix that begins: “In this appendix we shall give a Hilbert space interpretation of K(X). This is of interest in connection with the theory of the index for elliptic operators.” A bounded operator is Fredholm if its kernel and cokernel are finite-dimensional, and dim Ker T − dim Coker T “is called the index of T”. A footnote on the same page reads: “These results have been obtained independently by K. Jänich (Bonn dissertation 1964).” CITED
Atiyah's first step is the pair of inequalities dim Ker TS ≤ dim Ker T + dim Ker S and dim Coker TS ≤ dim Coker T + dim Coker S, “and so TS is again a Fredholm operator” (p. 154). The text layer of the scan reads ≤ as “<” and “dim” as “dirn”, and the script matches the passage in that damaged form. CITED
FredholmSpace.rank_ker_comp_le proves the kernel inequality for any two linear maps. It is stated for Mathlib's cardinal-valued rank, so it assumes no finiteness at all. The proof is Atiyah's: S carries the kernel of TS into the kernel of T, and what it loses on the way lies in the kernel of S. KERNEL-CHECKED The script checks both inequalities, and the index law Ind TS = Ind T + Ind S of Chapter 2, on 400 random pairs of matrices over ℚ. All 400 pass. COMPUTED
Atiyah's Theorem A1 says that index: [X, 𝔅] → K(X) is an isomorphism, where [X, 𝔅] is the set of homotopy classes of maps from a compact space X into the Fredholm operators. He then notes what this says when X is a point: the connected components of 𝔅 are labelled by an integer, and “this is in fact the index which explains our use of the word in the more general context”. Theorem A2 gives a second classifying space, the invertible elements of the quotient of all bounded operators by the compact ones, which is the Calkin algebra of Chapter 3. CITED
Noether's paper of 1921 (Chapter 3 §4) used “Index” for the winding number of a symbol. In Atiyah's usage it names the integer that labels a component. Between the two, the word moved from the symbol to the operator. Every integer does label something: every_integer_is_an_index gives each n ∈ ℤ as the index of Chapter 1's backward shift Bn (for n ≥ 0) or forward shift S|n| (for n < 0). KERNEL-CHECKED
Zois names the problem this section is about. For a family Ty one wants the index to be [Ker Ty] − [Coker Ty] as bundles over Y, but “Unfortunately these might not be vector bundles since the dimensions may jump.” The smallest example is a family of 2×3 matrices, T(t) = [[t, 0, 0], [0, 1, 0]]. The script computes it exactly: COMPUTED
| t | dim Ker | dim Coker | index | dim H/V − dim H′/T(V) |
|---|---|---|---|---|
| −2 | 1 | 0 | 1 | 1 |
| −½ | 1 | 0 | 1 | 1 |
| 0 | 2 | 1 | 1 | 1 |
| ⅓ | 1 | 0 | 1 | 1 |
| 5 | 1 | 0 | 1 | 1 |
At t = 0 the kernel and the cokernel each gain a dimension, and their difference does not change. Atiyah's remedy is to pick one subspace V, of finite codimension, that meets every kernel in 0. Here V = span(e₂) works. He then sets index T = [H/V] − [H/T(V)], a difference of two bundles that do not jump, and shows the answer does not depend on V (p. 158). In the table that is the last column: 1 at every t. A formal difference of two bundles is exactly an element of K(X), and that is how the index lands in K-theory. COMPUTED
To prove injectivity, Zois needs two facts. The first is that the invertible operators on Hilbert space form a contractible space, “The group GLH is contractible (this is difficult to prove)”. The second is the Whitehead Lemma, that diag(U, U⁻¹) can be joined to the identity through invertible matrices. CITED The second is easy to see. Let R(θ) be the block rotation. The path diag(U, 1)·R(θ)·diag(1, U⁻¹)·R(θ)⁻¹, for θ from 0 to π/2, starts at diag(U, U⁻¹) and ends at the identity. Its determinant is 1 all the way along. The script follows 20 random U ∈ GL₃(ℂ) at 201 points each and confirms this. COMPUTED