HomeTheorem List (p. 321 of 505)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 321 of 505) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-31179) |
Hilbert Space Explorer
(31180-32702) |
Users' Mathboxes
(32703-50434) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hosval 32001 | 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 32002 | 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 32003 | 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 32004 | 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 32005 | 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 32006 | 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 32007 | 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 32008 | 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 32009 | Define the Hilbert space zero operator. See df0op2 32013 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop = (projℎ‘0ℋ) | ||
| Definition | df-iop 32010 | Define the Hilbert space identity operator. See dfiop2 32014 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
| ⊢ Iop = (projℎ‘ ℋ) | ||
| Theorem | ho0val 32011 | 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 32012 | Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 0hop : ℋ⟶ ℋ | ||
| Theorem | df0op2 32013 | Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.) |
| ⊢ 0hop = ( ℋ × 0ℋ) | ||
| Theorem | dfiop2 32014 | Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.) |
| ⊢ Iop = ( I ↾ ℋ) | ||
| Theorem | hoif 32015 | 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 32016 | 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 32017 | 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 32018 | 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 32019 | 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 32020 | 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 32021* | 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 32022* | 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 32023 | 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 32024 | 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 32025 | 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 32026 | 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 32027 | 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 32028 | 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 32029 | 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 32030 | 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 32031 | 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 32032 | 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 32033 | 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 32034 | 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 32035 | 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 32036 | 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 32037 | 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 32038 | 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 32039 | 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 32040 | 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 32041 | 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 32042 | 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 32043 | 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 32044 | 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 32045 | 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 32046 | 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 32047 | 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 32048 | 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 32049 | 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 32050 | 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 32051 | 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 32052 | 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 32053 | 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 32054 | A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) | ||
| Theorem | hodseqi 32055 | 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 32056 | 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 32057 | 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 32058 | 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 32059 | 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 32060 | 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 32061 | 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 32062 | 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 32063 | 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 32064 | 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 32065 | 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 32066 | 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 32067 | 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 32068 | 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 32069 | 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 32070 | 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 32071 | 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 32072 | 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 32073 | 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 32074 | 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 32075 | 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 32076 | 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 32077 | 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 32078 | 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 32079 | 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 32080 | 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 32081 | 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 32082 | 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 32083 | 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 32084 | 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 32085 | 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 32086 | 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 32087 | 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 32088 | 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 32089* | 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 32090* | 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 32091* | 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 32092* | 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 32093* | 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 32094* | 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 32095 | 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 32096 | 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 32097 | 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 32098 | 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 32099 | 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)) | ||
| Definition | df-nmop 32100* | Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
| ⊢ normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < )) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |