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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjch1 27701 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((proj‘ ℋ)‘𝐴) = 𝐴)
 
Theorempjo 27702 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (((proj‘ ℋ)‘𝐴) − ((proj𝐻)‘𝐴)))
 
Theorempjcompi 27703 Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)
 
Theorempjidmi 27704 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘((proj𝐻)‘𝐴)) = ((proj𝐻)‘𝐴)
 
Theorempjadjii 27705 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵))
 
Theorempjaddii 27706 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵))
 
Theorempjinormii 27707 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2)
 
Theorempjmulii 27708 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐶 ∈ ℂ       ((proj𝐻)‘(𝐶 · 𝐴)) = (𝐶 · ((proj𝐻)‘𝐴))
 
Theorempjsubii 27709 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵))
 
Theorempjsslem 27710 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (((proj‘(⊥‘𝐻))‘𝐴) − ((proj‘(⊥‘𝐺))‘𝐴)) = (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴))
 
Theorempjss2i 27711 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → ((proj𝐻)‘((proj𝐺)‘𝐴)) = ((proj𝐻)‘𝐴))
 
Theorempjssmii 27712 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))
 
Theorempjssge0ii 27713 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴))
 
Theorempjdifnormii 27714 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴)))
 
Theorempjcji 27715 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))
 
Theorempjadji 27716 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵)))
 
Theorempjaddi 27717 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵)))
 
Theorempjinormi 27718 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2))
 
Theorempjsubi 27719 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵)))
 
Theorempjmuli 27720 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 · 𝐵)) = (𝐴 · ((proj𝐻)‘𝐵)))
 
Theorempjige0i 27721 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))
 
Theorempjige0 27722 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))
 
Theorempjcjt2 27723 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴))))
 
Theorempj0i 27724 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C       ((proj𝐻)‘0) = 0
 
Theorempjch 27725 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴))
 
Theorempjid 27726 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴𝐻) → ((proj𝐻)‘𝐴) = 𝐴)
 
Theorempjvec 27727* The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
(𝐻C𝐻 = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 𝑥})
 
Theorempjocvec 27728* The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 0})
 
Theorempjocini 27729 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (⊥‘(𝐺𝐻)) → ((proj𝐺)‘𝐴) ∈ (⊥‘(𝐺𝐻)))
 
Theorempjini 27730 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → ((proj𝐺)‘𝐴) ∈ (𝐺𝐻))
 
Theorempjjsi 27731* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)
𝐺C    &   𝐻S       (∀𝑥 ∈ (𝐺 𝐻)((proj‘(⊥‘𝐺))‘𝑥) ∈ 𝐻 → (𝐺 𝐻) = (𝐺 + 𝐻))
 
Theorempjfni 27732 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝐻C       (proj𝐻) Fn ℋ
 
Theorempjrni 27733 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝐻C       ran (proj𝐻) = 𝐻
 
Theorempjfoi 27734 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ–onto𝐻
 
Theorempjfi 27735 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ⟶ ℋ
 
Theorempjvi 27736 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)
 
Theorempjhfo 27737 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ–onto𝐻)
 
Theorempjrn 27738 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ran (proj𝐻) = 𝐻)
 
Theorempjhf 27739 The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ⟶ ℋ)
 
Theorempjfn 27740 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) Fn ℋ)
 
Theorempjsumi 27741 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((proj‘(𝐺 + 𝐻))‘𝐴) = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴))))
 
Theorempj11i 27742 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻)
 
Theorempjdsi 27743 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((𝐴 ∈ (𝐺 𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻)) → 𝐴 = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴)))
 
Theorempjds3i 27744 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((𝐴 ∈ ((𝐹 𝐺) ∨ 𝐻) ∧ 𝐹 ⊆ (⊥‘𝐺)) ∧ (𝐹 ⊆ (⊥‘𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻))) → 𝐴 = ((((proj𝐹)‘𝐴) + ((proj𝐺)‘𝐴)) + ((proj𝐻)‘𝐴)))
 
Theorempj11 27745 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
((𝐺C𝐻C ) → ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻))
 
Theorempjmfn 27746 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
proj Fn C
 
Theorempjmf1 27747 The projector function maps one-to-one into the set of Hilbert space operators. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
proj: C1-1→( ℋ ↑𝑚 ℋ)
 
Theorempjoi0 27748 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(((𝐺C𝐻C𝐴 ∈ ℋ) ∧ 𝐺 ⊆ (⊥‘𝐻)) → (((proj𝐺)‘𝐴) ·ih ((proj𝐻)‘𝐴)) = 0)
 
Theorempjoi0i 27749 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐺C    &   𝐻C    &   𝐴 ∈ ℋ       (𝐺 ⊆ (⊥‘𝐻) → (((proj𝐺)‘𝐴) ·ih ((proj𝐻)‘𝐴)) = 0)
 
Theorempjopythi 27750 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐺C    &   𝐻C    &   𝐴 ∈ ℋ       (𝐺 ⊆ (⊥‘𝐻) → ((norm‘(((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴)))↑2) = (((norm‘((proj𝐺)‘𝐴))↑2) + ((norm‘((proj𝐻)‘𝐴))↑2)))
 
Theorempjopyth 27751 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((norm‘(((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj𝐺)‘𝐴))↑2))))
 
Theorempjnormi 27752 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴)
 
Theorempjpythi 27753 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2))
 
Theorempjneli 27754 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴))
 
Theorempjnorm 27755 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴))
 
Theorempjpyth 27756 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2)))
 
Theorempjnel 27757 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (¬ 𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴)))
 
Theorempjnorm2 27758 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 27725 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) = (norm𝐴)))
 
19.5.11  Mayet's equation E_3
 
Theoremmayete3i 27759 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       (𝑋𝑌) ⊆ 𝑍
 
Theoremmayetes3i 27760 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑋)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       ((𝑋 𝑅) ∩ 𝑌) ⊆ (𝑍 𝑅)
 
19.6  Operators on Hilbert spaces
 
19.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from to . This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 28069.

 
Definitiondf-hosum 27761* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
+op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
 
Definitiondf-homul 27762* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
 
Definitiondf-hodif 27763* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
 
Definitiondf-hfsum 27764* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from to , not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
+fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
 
Definitiondf-hfmul 27765* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
 
Theoremhosmval 27766* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
 
Theoremhommval 27767* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
 
Theoremhodmval 27768* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))
 
Theoremhfsmval 27769* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
 
Theoremhfmmval 27770* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))
 
Theoremhosval 27771 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
 
Theoremhomval 27772 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))
 
Theoremhodval 27773 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) = ((𝑆𝐴) − (𝑇𝐴)))
 
Theoremhfsval 27774 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
 
Theoremhfmval 27775 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))
 
Theoremhoscl 27776 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ)
 
Theoremhomcl 27777 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) ∈ ℋ)
 
Theoremhodcl 27778 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) ∈ ℋ)
 
19.6.2  Zero and identity operators
 
Definitiondf-h0op 27779 Define the Hilbert space zero operator. See df0op2 27783 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop = (proj‘0)
 
Definitiondf-iop 27780 Define the Hilbert space identity operator. See dfiop2 27784 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Iop = (proj‘ ℋ)
 
Theoremho0val 27781 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( 0hop𝐴) = 0)
 
Theoremho0f 27782 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
0hop : ℋ⟶ ℋ
 
Theoremdf0op2 27783 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
0hop = ( ℋ × 0)
 
Theoremdfiop2 27784 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
Iop = ( I ↾ ℋ)
 
Theoremhoif 27785 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
Iop : ℋ–1-1-onto→ ℋ
 
Theoremhoival 27786 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( Iop𝐴) = 𝐴)
 
Theoremhoico1 27787 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∘ Iop ) = 𝑇)
 
Theoremhoico2 27788 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ( Iop𝑇) = 𝑇)
 
19.6.3  Operations on Hilbert space operators
 
Theoremhoaddcl 27789 The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
 
Theoremhomulcl 27790 The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
 
Theoremhoeq 27791* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑈𝑥) ↔ 𝑇 = 𝑈))
 
Theoremhoeqi 27792* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (∀𝑥 ∈ ℋ (𝑆𝑥) = (𝑇𝑥) ↔ 𝑆 = 𝑇)
 
Theoremhoscli 27793 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ)
 
Theoremhodcli 27794 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆op 𝑇)‘𝐴) ∈ ℋ)
 
Theoremhocoi 27795 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
 
Theoremhococli 27796 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) ∈ ℋ)
 
Theoremhocofi 27797 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆𝑇): ℋ⟶ ℋ
 
Theoremhocofni 27798 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆𝑇) Fn ℋ
 
Theoremhoaddcli 27799 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op 𝑇): ℋ⟶ ℋ
 
Theoremhosubcli 27800 Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇): ℋ⟶ ℋ
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