Compact perturbations of surjections

The functional-analytic Schwartz lemma used in the finite-dimensionality proof for cohomology on compact Riemann surfaces: a compact perturbation of a surjective map between Banach spaces has closed range of finite codimension. The kernel of the surjection need not be finite or complemented.

The proof uses the compact-operator quotient estimate and bounded nonlinear right inverses. These lower-level APIs require neither Fredholm hypotheses nor complemented kernels. No approximation property or finite-rank operator-norm approximation is assumed.

References

• O. Forster, Lectures on Riemann Surfaces, GTM 81, §14.

theorem ContinuousLinearMap.cofg_range_add_of_surjective_of_isCompactOperator {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (T K : E →L[𝕜] F) (hT : Function.Surjective ⇑T) (hK : IsCompactOperator ⇑K) : (↑(T + K)).range.CoFG

Schwartz's lemma. A compact perturbation of a surjection between real or complex Banach spaces has finite-dimensional cokernel, without any hypothesis on its kernel.

theorem ContinuousLinearMap.isClosed_range_add_of_surjective_of_isCompactOperator {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (T K : E →L[𝕜] F) (hT : Function.Surjective ⇑T) (hK : IsCompactOperator ⇑K) : IsClosed ↑(↑(T + K)).range

A compact perturbation of a surjection between real or complex Banach spaces has closed range. The finite-dimensional cokernel, rather than a kernel complement, supplies closedness.