feat(Analysis/Normed/Algebra/SpectralNorm): extensions of absolute values

[tb65536]

Let L be a finite extension of a complete normed field K. Then there is a unique absolute value on L extending the absolute value on K. This PR gives a construction that works uniformly across the archimedean and non-archimedean cases (bypassing Gelfand-Mazur).

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• depends on: feat(Analysis/Normed/Algebra/SpectralRadiusLimit): define the spectral radius limit lim ‖a ^ k‖ ^ (1 / k) #42607
• depends on: [Merged by Bors] - feat(Topology/Separation/Hausdorff): add tendsto_nhds_unique_of_forall #42986
• depends on: [Merged by Bors] - feat(Algebra/Group/Commute/Basic): an element commutes with its inverse #42987
• depends on: refactor(Analysis/Normed/Unbundled/SpectralNorm): extract, generalize, and golf uniqueness of power-multiplicative norms #43015
• depends on: [Merged by Bors] - feat(Analysis/Matrix/Normed): norm of block diagonal matrix #43016
• depends on: [Merged by Bors] - refactor(Analysis/Normed/Unbundled/SpectralNorm): extract, generalize, and golf algNormFromConst #43019
• depends on: chore(Algebra/Order/Hom/Basic): split file by imports #43109
• depends on: [Merged by Bors] - chore(Analysis/Normed/Unbundled/RingSeminorm): add missing API #43111

[github-actions]

PR summary 20c0f77e51

Import changes exceeding 2%

%	File
+2.52%	Mathlib.Analysis.Normed.Unbundled.SpectralNorm
+47.49%	Mathlib.Analysis.Subadditive

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Analysis.Subadditive	1333	1966	+633 (+47.49%)
Mathlib.Analysis.Normed.Unbundled.SpectralNorm	2456	2518	+62 (+2.52%)

Import changes for all files

Files	Import difference
27 files Mathlib.Analysis.CStarAlgebra.ApproximateUnit Mathlib.Analysis.CStarAlgebra.CStarMatrix Mathlib.Analysis.CStarAlgebra.Commutative.PosPart Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order Mathlib.Analysis.CStarAlgebra.Extreme Mathlib.Analysis.CStarAlgebra.GelfandDuality Mathlib.Analysis.CStarAlgebra.GelfandNaimarkSegal Mathlib.Analysis.CStarAlgebra.Hom Mathlib.Analysis.CStarAlgebra.Module.Constructions Mathlib.Analysis.CStarAlgebra.Module.Defs Mathlib.Analysis.CStarAlgebra.PositiveLinearFunctional Mathlib.Analysis.CStarAlgebra.PositiveLinearMap Mathlib.Analysis.CStarAlgebra.Projection Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart Mathlib.Analysis.CStarAlgebra.Spectrum Mathlib.Analysis.CStarAlgebra.Unitary.Connected Mathlib.Analysis.CStarAlgebra.Unitary.Span Mathlib.Analysis.InnerProductSpace.StandardSubspace Mathlib.Analysis.InnerProductSpace.StarOrder Mathlib.Analysis.Normed.Algebra.GelfandFormula Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder Mathlib.MeasureTheory.Measure.ResolventTransform	2
Mathlib.NumberTheory.Padics.Complex	58
3 files Mathlib.Analysis.Normed.Field.Dense Mathlib.Analysis.Normed.Field.Krasner Mathlib.Analysis.Normed.Unbundled.SpectralNorm	62
Mathlib.Analysis.Subadditive	633
Mathlib.Analysis.Normed.Algebra.SpectralRadiusLimit (new file)	1972
Mathlib.Analysis.Normed.Unbundled.IsPowMulUnique (new file)	2422
Mathlib.Analysis.Normed.Algebra.SpectralNorm (new file)	2517

---

Declarations diff (regex)

+ AlgebraNorm.copy
+ AlgebraNorm.eq_one_of_trivial
+ AlgebraNorm.le_one_of_trivial
+ AlgebraNorm.ringNorm
+ AlgebraNorm.toMulAlgebraNorm
+ Commute.spectralRadiusLim_add_le
+ Commute.spectralRadiusLim_mul_le
+ FiniteDimensional.spectralNorm_eq
+ Finset.map_prod_le_prod
+ IsAlgebraic.spectralNorm_eq
+ IsPowMul.unique
+ Matrix.spectralRadiusLim_blockMatrix
+ Matrix.spectralRadiusLim_conj
+ Matrix.spectralRadiusLim_conj_le
+ MulAlgebraNorm.unique
+ NormedAlgebra.norm_eq_spectralNorm'
+ Submultiplicative
+ eq_zero_of_spectralNorm_eq_zero
+ exists_le_spectralRadiusLim
+ exists_spectralRadiusLim_le
+ extension
+ extension_def
+ foo
+ foobar
+ instance (f : AlgebraNorm K L) (x : L) : Algebra (f.copy x) L
+ instance (f : AlgebraNorm K L) (x : L) : FiniteDimensional K (f.copy x)
+ instance (f : AlgebraNorm K L) (x : L) : NormedAlgebra K (f.copy x)
+ instance (f : AlgebraNorm K L) (x : L) : NormedRing (f.copy x)
+ instance : (v.extension L).LiesOver v
+ isNonarchimedean_spectralNorm'
+ isNonarchimedean_spectralRadiusLimNorm
+ isPowMul_spectralAlgNorm
+ isPowMul_spectralNorm'
+ isPowMul_spectralRadiusLimNorm
+ lim
+ norm_le_spectralNorm'
+ one_le_map_one
+ spectralAbsoluteValue
+ spectralAlgNorm'
+ spectralAlgNorm'_def
+ spectralAlgNorm'_extends
+ spectralAlgNorm'_isPowMul
+ spectralAlgNorm'_one
+ spectralMulAlgNorm'
+ spectralNorm'
+ spectralNorm'.eq_of_tower
+ spectralNorm'_eq_of_equiv
+ spectralNorm'_extends
+ spectralNorm'_mul
+ spectralNorm'_neg
+ spectralNorm'_one
+ spectralNorm'_smul
+ spectralNorm'_zero
+ spectralNorm.nontriviallyNormedField
+ spectralNorm.normedAlgebra
+ spectralNorm.normedAlgebra'
+ spectralNorm.normedField
+ spectralNorm_add
+ spectralNorm_algHom
+ spectralNorm_algebraMap
+ spectralNorm_eq_of_mem
+ spectralNorm_eq_zero
+ spectralRadiusLim
+ spectralRadiusLimNorm
+ spectralRadiusLimNorm_algebraMap
+ spectralRadiusLimNorm_apply
+ spectralRadiusLimNorm_def
+ spectralRadiusLimNorm_extends
+ spectralRadiusLimNorm_one
+ spectralRadiusLim_add_le
+ spectralRadiusLim_le_norm
+ spectralRadiusLim_le_norm_pow
+ spectralRadiusLim_mul_le
+ spectralRadiusLim_neg
+ spectralRadiusLim_nonneg
+ spectralRadiusLim_one
+ spectralRadiusLim_pow
+ spectralRadiusLim_pow_of_ne_zero
+ spectralRadiusLim_smul
+ spectralRadiusLim_zero
+ spectralRadius_eq_spectralRadiusLimit
+ tendsto_atBot
+ tendsto_lim
+ tendsto_spectralRadiusLim
+ tendsto_spectralRadiusLim'
- Subadditive
- norm_le_spectralNorm
- spectralAlgNorm_isPowMul

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 -- pending)

Computed after the build finishes.

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type (weak)
exposed public sections	5061	2

Current commit 20c0f77e51
Reference commit a120a09d6f

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.py 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).

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(Analysis/Normed/Algebra/SpectralRadiusLimit): define the spectral radius limit lim ‖a ^ k‖ ^ (1 / k) #42607
• [Merged by Bors] - feat(Topology/Separation/Hausdorff): add tendsto_nhds_unique_of_forall #42986
• [Merged by Bors] - feat(Algebra/Group/Commute/Basic): an element commutes with its inverse #42987
• refactor(Analysis/Normed/Unbundled/SpectralNorm): extract, generalize, and golf uniqueness of power-multiplicative norms #43015
• [Merged by Bors] - feat(Analysis/Matrix/Normed): norm of block diagonal matrix #43016
• [Merged by Bors] - refactor(Analysis/Normed/Unbundled/SpectralNorm): extract, generalize, and golf algNormFromConst #43019
• chore(Algebra/Order/Hom/Basic): split file by imports #43109
• [Merged by Bors] - chore(Analysis/Normed/Unbundled/RingSeminorm): add missing API #43111
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.