feat(Analysis/InnerProductSpace): inner products on exterior powers

[justus-springer]

Given a real inner product space E, we construct a canonical inner product on ⋀[ℝ]^n E via the Gram determinant formula: on decomposable elements, ⟪v₁ ∧ ⋯ ∧ vₙ, w₁ ∧ ⋯ ∧ wₙ⟫ = det (⟪vⱼ, wᵢ⟫)ᵢⱼ.

There are two generalizations of this construction: Going from ℝ to RCLike, and getting rid of the FiniteDimensional assumption. Both would require some prerequisites, hence they are left as future work for now (see the future work section of the module docstring).

---

[github-actions]

PR summary 1935330e30

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.InnerProductSpace.ExteriorPower (new file)	2562

---

Declarations diff (regex)

+ OrthonormalBasis.exteriorPower
+ OrthonormalBasis.toBasis_exteriorPower
+ flip_innerProductForm
+ gram_eq_one_iff_orthonormal
+ innerProductForm
+ innerProductForm_eq_sum
+ innerProductForm_self
+ innerProductForm_symm
+ innerProductForm_ιMulti_family_of_orthonormal
+ innerProductForm_ιMulti_self
+ innerProductForm_ιMulti_ιMulti
+ inner_ιMulti_self
+ inner_ιMulti_ιMulti
+ instance [FiniteDimensional ℝ E] : InnerProductSpace ℝ (⋀[ℝ]^n E)
+ instance [FiniteDimensional ℝ E] : InnerProductSpace.Core ℝ (⋀[ℝ]^n E)
+ instance [FiniteDimensional ℝ E] : NormedAddCommGroup (⋀[ℝ]^n E)

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 1935330).

• +8 new declarations
• −0 removed declarations

+Matrix.gram_eq_one_iff_orthonormal
+OrthonormalBasis.exteriorPower
+OrthonormalBasis.toBasis_exteriorPower
+exteriorPower.inner_ιMulti_self
+exteriorPower.inner_ιMulti_ιMulti
+exteriorPower.instCoreRealSubtypeExteriorAlgebraMemSubmoduleExteriorPowerOfFiniteDimensional
+exteriorPower.instInnerProductSpaceRealSubtypeExteriorAlgebraMemSubmoduleExteriorPower
+exteriorPower.instNormedAddCommGroupSubtypeExteriorAlgebraRealMemSubmoduleExteriorPowerOfFiniteDimensional

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5021	1	exposed public sections

Current commit 1935330e30
Reference commit 65fe2a1f85

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).

[github-actions]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions.
Thank you!

[tb65536]

Thanks!

maintainer delegate

[github-actions]

🚀 Pull request has been placed on the maintainer queue by tb65536.

[riccardobrasca]

Thanks!

bors d+

[riccardobrasca]

What about not making this public?

[justus-springer]

For some reason I thought this wasn't possible, but now I see it is if I put @[no_expose] on the inner product instance. In that case, yes I think that's better.

[mathlib-bors]

✌️ justus-springer can now approve this pull request until 2026-08-10 05:44 UTC (in 2 weeks). To approve and merge, reply with bors r+. More detailed instructions are available here.

⚠️ This delegation only covers changes within Archive/**, Counterexamples/**, docs/**, DownstreamTest/**, Mathlib/**, MathlibTest/**, widget/**, Archive.lean, Counterexamples.lean, docs.lean, Mathlib.lean; an author commit touching anything else will revoke it. Bors also revokes it if a later push changes too many files for it to check the full list — even if it stays within scope.

[tb65536]

@justus-springer The build was failing due to long lines. I've pushed a fix.