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Rows of diagonal tensor maps

Lax342547.ComponentDuals · concepts/Lax342547/ComponentDuals.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Dualizing a componentwise tensor map preserves its componentwise row spaces through the finite product dual equivalence; these row cover spaces are closed under joins.

    Concept map
    23 concepts
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 diagonal_dual_identity proven

    2 diagonal_dual_range proven

    3 dual_componentwise_join_closed proven

    Lean source view on GitHub

    1import Lax342547.ComponentSpaces
    2
    3/-!
    4---
    5title: Rows of diagonal tensor maps
    6type: lemma
    7---
    8Dualizing a componentwise tensor map preserves its componentwise row spaces through the finite product dual equivalence; these row cover spaces are closed under joins.
    9-/
    10
    11namespace Lax342547.ComponentDuals
    12
    13open Lax342547.ComponentSpaces Lax342547.CoverClasses
    14open scoped BigOperators
    15
    16def DualComponentwise {K e : Type} [Field K] [Fintype e] [DecidableEq e] {V : e → Type}
    17 [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    18 (T : Submodule K (Module.Dual K (∀ i, V i))) : Prop :=
    19 Componentwise (T.map (LinearMap.lsum K V K).symm.toLinearMap)
    20
    21axiom dual_componentwise_join_closed {K e : Type} [Field K] [Fintype e] [DecidableEq e]
    22 {V : e → Type} [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)] :
    23 JoinClosed (DualComponentwise (K := K) (V := V))
    24
    25axiom diagonal_dual_identity {K e : Type} [Field K] [Fintype e] [DecidableEq e]
    26 {V W : e → Type} [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    27 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)] (M : ∀ i, V i →ₗ[K] W i) :
    28 (LinearMap.piMap M).dualMap.comp (LinearMap.lsum K W K).toLinearMap =
    29 (LinearMap.lsum K V K).toLinearMap.comp (LinearMap.piMap (fun i => (M i).dualMap))
    30
    31axiom diagonal_dual_range {K e : Type} [Field K] [Fintype e] [DecidableEq e]
    32 {V W : e → Type} [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    33 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)] (M : ∀ i, V i →ₗ[K] W i) :
    34 (LinearMap.range (LinearMap.piMap M).dualMap).map (LinearMap.lsum K V K).symm.toLinearMap =
    35 coordinateSpace (fun i => LinearMap.range (M i).dualMap)
    36
    37end Lax342547.ComponentDuals
    38
    Show ProofShow ProofShow Proof
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