feat(Analysis/InnerProductSpace/Adjoint): characterize least-squares minimizers

[bocowgill]

This PR adds two theorems about least-squares minimizers for a continuous linear map A.

The first theorem states that the fitted value A x minimizes the distance to y over A.range if and only if the adjoint maps the residual y - A x to zero.

The second theorem restates the result in coefficient-space. That is, x minimizes ‖y - A z‖ over all z : E if and only if the same adjoint condition holds. This result is useful because it makes the range-level characterization applicable for least-squares problems (which are often stated in terms of coefficients).

These results provide a characterization of least-squares minimizers. The theorems use the existing APIs for range, orthogonal-complement, and adjoints. The PR adds them to Mathlib/Analysis/InnerProductSpace/Adjoint.lean.

These results were added to Mathlib/Analysis/InnerProductSpace/Adjoint.lean because a least-squares residual is orthogonal to A.range, while the orthogonal complement of A.range is the kernel of the adjoint.

---

LLM disclosure

For theorem planning and Lean implementation, I used OpenAI Codex substantially. I also used it for suggestions around naming and documentation, for iterating on the proofs, and for validating my commits locally.

I manually searched for the best locations and earlier results to build upon. I personally reviewed both the mathematical statements and Lean code (line-by-line), as well as the suggestions around naming and API choices. I also reviewed and confirmed that I can explain each step of the two theorems' proofs.

To validate the change, I directly compiled the modified file and ran a targeted build of Mathlib.Analysis.InnerProductSpace.Adjoint. Then I ran the style linter, and completed a full lake build and a full lake test.

Please note: This is my first Mathlib contribution. I welcome corrections or suggestions about all aspects of this PR. I'm prepared to revise it as necessary, and to learn from the review.

[github-actions]

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[bocowgill]

LLM-generated

[github-actions]

PR summary 322cf58a29

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ forall_norm_sub_apply_le_iff_adjoint_apply_sub_eq_zero
+ norm_sub_apply_eq_iInf_iff_adjoint_apply_sub_eq_zero

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 322cf58).

• +2 new declarations
• −0 removed declarations

+ContinuousLinearMap.forall_norm_sub_apply_le_iff_adjoint_apply_sub_eq_zero
+ContinuousLinearMap.norm_sub_apply_eq_iInf_iff_adjoint_apply_sub_eq_zero

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 322cf58a29
Reference commit 6996953ff1

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).