HomeTheorem List (p. 318 of 498)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hfmmval 31701* | 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 · (𝑇‘𝑥)))) | ||
| Theorem | hosval 31702 | 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) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) | ||
| Theorem | homval 31703 | 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 ·ℎ (𝑇‘𝐵))) | ||
| Theorem | hodval 31704 | 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) → ((𝑆 −op 𝑇)‘𝐴) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) | ||
| Theorem | hfsval 31705 | 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) = ((𝑆‘𝐴) + (𝑇‘𝐴))) | ||
| Theorem | hfmval 31706 | 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 · (𝑇‘𝐵))) | ||
| Theorem | hoscl 31707 | 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) | ||
| Theorem | homcl 31708 | 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) | ||
| Theorem | hodcl 31709 | 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) → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) | ||
| Definition | df-h0op 31710 | Define the Hilbert space zero operator. See df0op2 31714 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop = (projℎ‘0ℋ) | ||
| Definition | df-iop 31711 | Define the Hilbert space identity operator. See dfiop2 31715 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ Iop = (projℎ‘ ℋ) | ||
| Theorem | ho0val 31712 | 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 → ( 0hop ‘𝐴) = 0ℎ) | ||
| Theorem | ho0f 31713 | Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 0hop : ℋ⟶ ℋ | ||
| Theorem | df0op2 31714 | Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.) |
| ⊢ 0hop = ( ℋ × 0ℋ) | ||
| Theorem | dfiop2 31715 | Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.) |
| ⊢ Iop = ( I ↾ ℋ) | ||
| Theorem | hoif 31716 | Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
| ⊢ Iop : ℋ–1-1-onto→ ℋ | ||
| Theorem | hoival 31717 | The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → ( Iop ‘𝐴) = 𝐴) | ||
| Theorem | hoico1 31718 | Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 ∘ Iop ) = 𝑇) | ||
| Theorem | hoico2 31719 | Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → ( Iop ∘ 𝑇) = 𝑇) | ||
| Theorem | hoaddcl 31720 | 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) | ||
| Theorem | homulcl 31721 | 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) | ||
| Theorem | hoeq 31722* | Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈)) | ||
| Theorem | hoeqi 31723* | Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥) ↔ 𝑆 = 𝑇) | ||
| Theorem | hoscli 31724 | 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) | ||
| Theorem | hodcli 31725 | Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((𝑆 −op 𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | hocoi 31726 | Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | ||
| Theorem | hococli 31727 | 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 → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | hocofi 31728 | Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ | ||
| Theorem | hocofni 31729 | Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 ∘ 𝑇) Fn ℋ | ||
| Theorem | hoaddcli 31730 | 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 | ||
| Theorem | hosubcli 31731 | 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 ⇒ ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ | ||
| Theorem | hoaddfni 31732 | 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 | ||
| Theorem | hosubfni 31733 | Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
| ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑆 −op 𝑇) Fn ℋ | ||
| Theorem | hoaddcomi 31734 | 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) | ||
| Theorem | hosubcl 31735 | Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇): ℋ⟶ ℋ) | ||
| Theorem | hoaddcom 31736 | 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)) | ||
| Theorem | hodsi 31737 | 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 ⇒ ⊢ ((𝑅 −op 𝑆) = 𝑇 ↔ (𝑆 +op 𝑇) = 𝑅) | ||
| Theorem | hoaddassi 31738 | 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)) | ||
| Theorem | hoadd12i 31739 | 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)) | ||
| Theorem | hoadd32i 31740 | 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) | ||
| Theorem | hocadddiri 31741 | 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) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)) | ||
| Theorem | hocsubdiri 31742 | 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 ⇒ ⊢ ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇)) | ||
| Theorem | ho2coi 31743 | 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 → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) | ||
| Theorem | hoaddass 31744 | 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))) | ||
| Theorem | hoadd32 31745 | 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)) | ||
| Theorem | hoadd4 31746 | 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))) | ||
| Theorem | hocsubdir 31747 | 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) → ((𝑅 −op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) −op (𝑆 ∘ 𝑇))) | ||
| Theorem | hoaddridi 31748 | 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 | ||
| Theorem | hodidi 31749 | Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑇 −op 𝑇) = 0hop | ||
| Theorem | ho0coi 31750 | Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ( 0hop ∘ 𝑇) = 0hop | ||
| Theorem | hoid1i 31751 | Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑇 ∘ Iop ) = 𝑇 | ||
| Theorem | hoid1ri 31752 | Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ( Iop ∘ 𝑇) = 𝑇 | ||
| Theorem | hoaddrid 31753 | 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) | ||
| Theorem | hodid 31754 | Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 𝑇) = 0hop ) | ||
| Theorem | hon0 31755 | A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) | ||
| Theorem | hodseqi 31756 | 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 (𝑇 −op 𝑆)) = 𝑇 | ||
| Theorem | ho0subi 31757 | 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 ⇒ ⊢ (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇)) | ||
| Theorem | honegsubi 31758 | 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)) = (𝑆 −op 𝑇) | ||
| Theorem | ho0sub 31759 | 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) → (𝑆 −op 𝑇) = (𝑆 +op ( 0hop −op 𝑇))) | ||
| Theorem | hosubid1 31760 | The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 −op 0hop ) = 𝑇) | ||
| Theorem | honegsub 31761 | 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)) = (𝑇 −op 𝑈)) | ||
| Theorem | homullid 31762 | 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) | ||
| Theorem | homco1 31763 | 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 (𝑇 ∘ 𝑈))) | ||
| Theorem | homulass 31764 | 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))) | ||
| Theorem | hoadddi 31765 | 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))) | ||
| Theorem | hoadddir 31766 | 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))) | ||
| Theorem | homul12 31767 | 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))) | ||
| Theorem | honegneg 31768 | 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) | ||
| Theorem | hosubneg 31769 | Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 −op (-1 ·op 𝑈)) = (𝑇 +op 𝑈)) | ||
| Theorem | hosubdi 31770 | 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 (𝑇 −op 𝑈)) = ((𝐴 ·op 𝑇) −op (𝐴 ·op 𝑈))) | ||
| Theorem | honegdi 31771 | 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))) | ||
| Theorem | honegsubdi 31772 | Distribution of negative over subtraction. (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 𝑈)) | ||
| Theorem | honegsubdi2 31773 | Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op (𝑇 −op 𝑈)) = (𝑈 −op 𝑇)) | ||
| Theorem | hosubsub2 31774 | 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) → (𝑆 −op (𝑇 −op 𝑈)) = (𝑆 +op (𝑈 −op 𝑇))) | ||
| Theorem | hosub4 31775 | 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)) = ((𝑅 −op 𝑇) +op (𝑆 −op 𝑈))) | ||
| Theorem | hosubadd4 31776 | 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)) = ((𝑅 +op 𝑈) −op (𝑆 +op 𝑇))) | ||
| Theorem | hoaddsubass 31777 | Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = (𝑆 +op (𝑇 −op 𝑈))) | ||
| Theorem | hoaddsub 31778 | Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 +op 𝑇) −op 𝑈) = ((𝑆 −op 𝑈) +op 𝑇)) | ||
| Theorem | hosubsub 31779 | Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑆 −op (𝑇 −op 𝑈)) = ((𝑆 −op 𝑇) +op 𝑈)) | ||
| Theorem | hosubsub4 31780 | Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝑆 −op 𝑇) −op 𝑈) = (𝑆 −op (𝑇 +op 𝑈))) | ||
| Theorem | ho2times 31781 | Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) | ||
| Theorem | hoaddsubassi 31782 | Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) −op 𝑇) = (𝑅 +op (𝑆 −op 𝑇)) | ||
| Theorem | hoaddsubi 31783 | Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 +op 𝑆) −op 𝑇) = ((𝑅 −op 𝑇) +op 𝑆) | ||
| Theorem | hosd1i 31784 | Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ (𝑇 +op 𝑈) = (𝑇 −op ( 0hop −op 𝑈)) | ||
| Theorem | hosd2i 31785 | Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ (𝑇 +op 𝑈) = (𝑇 −op ((𝑈 −op 𝑈) −op 𝑈)) | ||
| Theorem | hopncani 31786 | Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 +op 𝑈) −op 𝑈) = 𝑇 | ||
| Theorem | honpcani 31787 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) +op 𝑈) = 𝑇 | ||
| Theorem | hosubeq0i 31788 | If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) | ||
| Theorem | honpncani 31789 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 −op 𝑆) +op (𝑆 −op 𝑇)) = (𝑅 −op 𝑇) | ||
| Theorem | ho01i 31790* | A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | ho02i 31791* | A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | hoeq1 31792* | A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇)) | ||
| Theorem | hoeq2 31793* | A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇)) | ||
| Theorem | adjmo 31794* | Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) | ||
| Theorem | adjsym 31795* | Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·ih 𝑦))) | ||
| Theorem | eigrei 31796 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
| Theorem | eigre 31797 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
| Theorem | eigposi 31798 | A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((((𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | ||
| Theorem | eigorthi 31799 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐷 ∈ ℂ ⇒ ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
| Theorem | eigorth 31800 | A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
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