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Theorem List for Metamath Proof Explorer - 28701-28800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempj11 28701 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
((𝐺C𝐻C ) → ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻))
 
Theorempjmfn 28702 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
proj Fn C
 
Theorempjmf1 28703 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 28704 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 28705 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 28706 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 28707 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 28708 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 28709 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 28710 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 28711 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 28712 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 28713 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 28714 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 28681 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 28715 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 28716 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 29025.

 
Definitiondf-hosum 28717* 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 28718* 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 28719* 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 28720* 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 28721* 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 28722* 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 28723* 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 28724* 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 28725* 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 28726* 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 28727 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 28728 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 28729 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 28730 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 28731 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 28732 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 28733 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 28734 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 28735 Define the Hilbert space zero operator. See df0op2 28739 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop = (proj‘0)
 
Definitiondf-iop 28736 Define the Hilbert space identity operator. See dfiop2 28740 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Iop = (proj‘ ℋ)
 
Theoremho0val 28737 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 28738 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
0hop : ℋ⟶ ℋ
 
Theoremdf0op2 28739 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
0hop = ( ℋ × 0)
 
Theoremdfiop2 28740 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
Iop = ( I ↾ ℋ)
 
Theoremhoif 28741 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
Iop : ℋ–1-1-onto→ ℋ
 
Theoremhoival 28742 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( Iop𝐴) = 𝐴)
 
Theoremhoico1 28743 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∘ Iop ) = 𝑇)
 
Theoremhoico2 28744 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 28745 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 28746 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 28747* 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 28748* 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 28749 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 28750 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 28751 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 28752 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 28753 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 28754 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 28755 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 28756 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
 
Theoremhoaddfni 28757 Functionality 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) Fn ℋ
 
Theoremhosubfni 28758 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇) Fn ℋ
 
Theoremhoaddcomi 28759 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op 𝑇) = (𝑇 +op 𝑆)
 
Theoremhosubcl 28760 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇): ℋ⟶ ℋ)
 
Theoremhoaddcom 28761 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑇 +op 𝑆))
 
Theoremhodsi 28762 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅)
 
Theoremhoaddassi 28763 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
 
Theoremhoadd12i 28764 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇))
 
Theoremhoadd32i 28765 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆)
 
Theoremhocadddiri 28766 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))
 
Theoremhocsubdiri 28767 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅op 𝑆) ∘ 𝑇) = ((𝑅𝑇) −op (𝑆𝑇))
 
Theoremho2coi 28768 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))
 
Theoremhoaddass 28769 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))
 
Theoremhoadd32 28770 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆))
 
Theoremhoadd4 28771 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) +op (𝑇 +op 𝑈)) = ((𝑅 +op 𝑇) +op (𝑆 +op 𝑈)))
 
Theoremhocsubdir 28772 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑅op 𝑆) ∘ 𝑇) = ((𝑅𝑇) −op (𝑆𝑇)))
 
Theoremhoaddid1i 28773 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇 +op 0hop ) = 𝑇
 
Theoremhodidi 28774 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇op 𝑇) = 0hop
 
Theoremho0coi 28775 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( 0hop𝑇) = 0hop
 
Theoremhoid1i 28776 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (𝑇 ∘ Iop ) = 𝑇
 
Theoremhoid1ri 28777 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       ( Iop𝑇) = 𝑇
 
Theoremhoaddid1 28778 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 +op 0hop ) = 𝑇)
 
Theoremhodid 28779 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 𝑇) = 0hop )
 
Theoremhon0 28780 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)
 
Theoremhodseqi 28781 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (𝑇op 𝑆)) = 𝑇
 
Theoremho0subi 28782 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇))
 
Theoremhonegsubi 28783 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑆 +op (-1 ·op 𝑇)) = (𝑆op 𝑇)
 
Theoremho0sub 28784 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑆 +op ( 0hopop 𝑇)))
 
Theoremhosubid1 28785 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇op 0hop ) = 𝑇)
 
Theoremhonegsub 28786 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op (-1 ·op 𝑈)) = (𝑇op 𝑈))
 
Theoremhomulid2 28787 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)
 
Theoremhomco1 28788 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) ∘ 𝑈) = (𝐴 ·op (𝑇𝑈)))
 
Theoremhomulass 28789 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 · 𝐵) ·op 𝑇) = (𝐴 ·op (𝐵 ·op 𝑇)))
 
Theoremhoadddi 28790 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))
 
Theoremhoadddir 28791 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))
 
Theoremhomul12 28792 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op (𝐵 ·op 𝑇)) = (𝐵 ·op (𝐴 ·op 𝑇)))
 
Theoremhonegneg 28793 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (-1 ·op (-1 ·op 𝑇)) = 𝑇)
 
Theoremhosubneg 28794 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇op (-1 ·op 𝑈)) = (𝑇 +op 𝑈))
 
Theoremhosubdi 28795 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈)))
 
Theoremhonegdi 28796 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 +op 𝑈)) = ((-1 ·op 𝑇) +op (-1 ·op 𝑈)))
 
Theoremhonegsubdi 28797 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = ((-1 ·op 𝑇) +op 𝑈))
 
Theoremhonegsubdi2 28798 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇op 𝑈)) = (𝑈op 𝑇))
 
Theoremhosubsub2 28799 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆op (𝑇op 𝑈)) = (𝑆 +op (𝑈op 𝑇)))
 
Theoremhosub4 28800 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝑅 +op 𝑆) −op (𝑇 +op 𝑈)) = ((𝑅op 𝑇) +op (𝑆op 𝑈)))
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