File: Mathlib/Analysis/InnerProductSpace/Positive.lean

full_name: ContinuousLinearMap.«_aux_Mathlib_Analysis_InnerProductSpace_Positive___macroRules_ContinuousLinearMap_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
goal: Lean.Macro

full_name: ContinuousLinearMap.IsPositive
code: /-- A continuous linear endomorphism `T` of a Hilbert space is **positive** if it is self adjoint
  and `∀ x, 0 ≤ re ⟪T x, x⟫`. -/
def IsPositive (T : E →L[𝕜] E) : Prop 
goal: Prop

full_name: ContinuousLinearMap.IsPositive.isSelfAdjoint
code: theorem IsPositive.isSelfAdjoint {T : E →L[𝕜] E} (hT : IsPositive T) : IsSelfAdjoint T 
goal: IsSelfAdjoint T

full_name: ContinuousLinearMap.IsPositive.inner_nonneg_left
code: theorem IsPositive.inner_nonneg_left {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
    0 ≤ re ⟪T x, x⟫ 
goal: 0 ≤ RCLike.re ⟪T x, x⟫_𝕜

full_name: ContinuousLinearMap.IsPositive.inner_nonneg_right
code: theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
    0 ≤ re ⟪x, T x⟫ 
goal: 0 ≤ RCLike.re ⟪x, T x⟫_𝕜

full_name: ContinuousLinearMap.isPositive_zero
code: theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) 
goal: ContinuousLinearMap.IsPositive 0

full_name: ContinuousLinearMap.isPositive_one
code: theorem isPositive_one : IsPositive (1 : E →L[𝕜] E) 
goal: ContinuousLinearMap.IsPositive 1

full_name: ContinuousLinearMap.IsPositive.add
code: theorem IsPositive.add {T S : E →L[𝕜] E} (hT : T.IsPositive) (hS : S.IsPositive) :
    (T + S).IsPositive 
goal: (T + S).IsPositive

full_name: ContinuousLinearMap.IsPositive.conj_adjoint
code: theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
    (S ∘L T ∘L S†).IsPositive 
goal: (S.comp (T.comp (ContinuousLinearMap.adjoint S))).IsPositive

full_name: ContinuousLinearMap.IsPositive.adjoint_conj
code: theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
    (S† ∘L T ∘L S).IsPositive 
goal: ((ContinuousLinearMap.adjoint S).comp (T.comp S)).IsPositive

full_name: ContinuousLinearMap.IsPositive.conj_orthogonalProjection
code: theorem IsPositive.conj_orthogonalProjection (U : Submodule 𝕜 E) {T : E →L[𝕜] E} (hT : T.IsPositive)
    [CompleteSpace U] :
    (U.subtypeL ∘L
        orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U).IsPositive 
goal: (U.subtypeL.comp ((orthogonalProjection U).comp (T.comp (U.subtypeL.comp (orthogonalProjection U))))).IsPositive

full_name: ContinuousLinearMap.IsPositive.orthogonalProjection_comp
code: theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPositive) (U : Submodule 𝕜 E)
    [CompleteSpace U] : (orthogonalProjection U ∘L T ∘L U.subtypeL).IsPositive 
goal: ((orthogonalProjection U).comp (T.comp U.subtypeL)).IsPositive

full_name: ContinuousLinearMap.isPositive_iff_complex
code: theorem isPositive_iff_complex (T : E' →L[ℂ] E') :
    IsPositive T ↔ ∀ x, (re ⟪T x, x⟫_ℂ : ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ re ⟪T x, x⟫_ℂ 
goal: T.IsPositive ↔ ∀ (x : E'), ↑(RCLike.re ⟪T x, x⟫_ℂ) = ⟪T x, x⟫_ℂ ∧ 0 ≤ RCLike.re ⟪T x, x⟫_ℂ

full_name: ContinuousLinearMap.instLoewnerPartialOrder
code: /-- The (Loewner) partial order on continuous linear maps on a Hilbert space determined by
`f ≤ g` if and only if `g - f` is a positive linear map (in the sense of
`ContinuousLinearMap.IsPositive`). With this partial order, the continuous linear maps form a
`StarOrderedRing`. -/
instance instLoewnerPartialOrder : PartialOrder (E →L[𝕜] E) 
goal: PartialOrder (E →L[𝕜] E)

full_name: ContinuousLinearMap.le_def
code: lemma le_def (f g : E →L[𝕜] E) : f ≤ g ↔ (g - f).IsPositive 
goal: f ≤ g ↔ (g - f).IsPositive

full_name: ContinuousLinearMap.nonneg_iff_isPositive
code: lemma nonneg_iff_isPositive (f : E →L[𝕜] E) : 0 ≤ f ↔ f.IsPositive 
goal: 0 ≤ f ↔ f.IsPositive