Documentation

Mathlib.Analysis.NormedSpace.MStructure

M-structure #

A projection P on a normed space X is said to be an L-projection (IsLprojection) if, for all x in X, x=Px+(1P)x.

A projection P on a normed space X is said to be an M-projection if, for all x in X, x=max(Px,(1P)x).

The L-projections on X form a Boolean algebra (IsLprojection.Subtype.BooleanAlgebra).

TODO (Motivational background) #

The M-projections on a normed space form a Boolean algebra.

The range of an L-projection on a normed space X is said to be an L-summand of X. The range of an M-projection is said to be an M-summand of X.

When X is a Banach space, the Boolean algebra of L-projections is complete. Let X be a normed space with dual X^*. A closed subspace M of X is said to be an M-ideal if the topological annihilator M^∘ is an L-summand of X^*.

M-ideal, M-summands and L-summands were introduced by Alfsen and Effros in [alfseneffros1972] to study the structure of general Banach spaces. When A is a JB*-triple, the M-ideals of A are exactly the norm-closed ideals of A. When A is a JBW*-triple with predual X, the M-summands of A are exactly the weak*-closed ideals, and their pre-duals can be identified with the L-summands of X. In the special case when A is a C*-algebra, the M-ideals are exactly the norm-closed two-sided ideals of A, when A is also a W*-algebra the M-summands are exactly the weak*-closed two-sided ideals of A.

Implementation notes #

The approach to showing that the L-projections form a Boolean algebra is inspired by MeasureTheory.MeasurableSpace.

Instead of using P : X →L[𝕜] X to represent projections, we use an arbitrary ring M with a faithful action on X. ContinuousLinearMap.apply_module can be used to recover the X →L[𝕜] X special case.

References #

Tags #

M-summand, M-projection, L-summand, L-projection, M-ideal, M-structure

structure IsLprojection (X : Type u_1) [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : M) :

A projection on a normed space X is said to be an L-projection if, for all x in X, x=Px+(1P)x.

Note that we write P • x instead of P x for reasons described in the module docstring.

Instances For
    theorem IsLprojection.proj {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsLprojection X P) :
    theorem IsLprojection.Lnorm {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsLprojection X P) (x : X) :
    x = P x + (1 - P) x
    structure IsMprojection (X : Type u_1) [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : M) :

    A projection on a normed space X is said to be an M-projection if, for all x in X, x=max(Px,(1P)x).

    Note that we write P • x instead of P x for reasons described in the module docstring.

    Instances For
      theorem IsMprojection.proj {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsMprojection X P) :
      theorem IsMprojection.Mnorm {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (self : IsMprojection X P) (x : X) :
      theorem IsLprojection.Lcomplement {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : M} (h : IsLprojection X P) :
      theorem IsLprojection.Lcomplement_iff {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : M) :
      theorem IsLprojection.commute {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P : M} {Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
      theorem IsLprojection.mul {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P : M} {Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
      theorem IsLprojection.join {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P : M} {Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
      IsLprojection X (P + Q - P * Q)
      instance IsLprojection.Subtype.hasCompl {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] :
      HasCompl { f : M // IsLprojection X f }
      Equations
      • IsLprojection.Subtype.hasCompl = { compl := fun (P : { f : M // IsLprojection X f }) => 1 - P, }
      @[simp]
      theorem IsLprojection.coe_compl {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] (P : { P : M // IsLprojection X P }) :
      P = 1 - P
      instance IsLprojection.Subtype.inf {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      Inf { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.inf = { inf := fun (P Q : { P : M // IsLprojection X P }) => P * Q, }
      @[simp]
      theorem IsLprojection.coe_inf {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P : { P : M // IsLprojection X P }) (Q : { P : M // IsLprojection X P }) :
      (P Q) = P * Q
      instance IsLprojection.Subtype.sup {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      Sup { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.sup = { sup := fun (P Q : { P : M // IsLprojection X P }) => P + Q - P * Q, }
      @[simp]
      theorem IsLprojection.coe_sup {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P : { P : M // IsLprojection X P }) (Q : { P : M // IsLprojection X P }) :
      (P Q) = P + Q - P * Q
      instance IsLprojection.Subtype.sdiff {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      SDiff { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.sdiff = { sdiff := fun (P Q : { P : M // IsLprojection X P }) => P * (1 - Q), }
      @[simp]
      theorem IsLprojection.coe_sdiff {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P : { P : M // IsLprojection X P }) (Q : { P : M // IsLprojection X P }) :
      (P \ Q) = P * (1 - Q)
      instance IsLprojection.Subtype.partialOrder {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      PartialOrder { P : M // IsLprojection X P }
      Equations
      theorem IsLprojection.le_def {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] (P : { P : M // IsLprojection X P }) (Q : { P : M // IsLprojection X P }) :
      P Q P = (P Q)
      instance IsLprojection.Subtype.zero {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] :
      Zero { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.zero = { zero := 0, }
      @[simp]
      theorem IsLprojection.coe_zero {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] :
      0 = 0
      instance IsLprojection.Subtype.one {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] :
      One { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.one = { one := 1, }
      @[simp]
      theorem IsLprojection.coe_one {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] :
      1 = 1
      instance IsLprojection.Subtype.boundedOrder {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      BoundedOrder { P : M // IsLprojection X P }
      Equations
      • IsLprojection.Subtype.boundedOrder = BoundedOrder.mk
      @[simp]
      theorem IsLprojection.coe_bot {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      = 0
      @[simp]
      theorem IsLprojection.coe_top {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      = 1
      theorem IsLprojection.compl_mul {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : { P : M // IsLprojection X P }} {Q : M} :
      P * Q = Q - P * Q
      theorem IsLprojection.mul_compl_self {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] {P : { P : M // IsLprojection X P }} :
      P * P = 0
      theorem IsLprojection.distrib_lattice_lemma {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] {P : { P : M // IsLprojection X P }} {Q : { P : M // IsLprojection X P }} {R : { P : M // IsLprojection X P }} :
      (P + P * R) * (P + Q * R * P) = P + Q * R * P
      Equations
      • IsLprojection.instLatticeSubtypeOfFaithfulSMul = Lattice.mk
      instance IsLprojection.Subtype.distribLattice {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      DistribLattice { P : M // IsLprojection X P }
      Equations
      instance IsLprojection.Subtype.BooleanAlgebra {X : Type u_1} [NormedAddCommGroup X] {M : Type u_2} [Ring M] [Module M X] [FaithfulSMul M X] :
      BooleanAlgebra { P : M // IsLprojection X P }
      Equations