HomeTheorem List (p. 322 of 504)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | kbfval 32101* | The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation. See df-kb 32000. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵) = (𝑥 ∈ ℋ ↦ ((𝑥 ·ih 𝐵) ·ℎ 𝐴))) | ||
| Theorem | kbop 32102 | The outer product of two vectors, expressed as ∣ 𝐴⟩⟨𝐵 ∣ in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ketbra 𝐵): ℋ⟶ ℋ) | ||
| Theorem | kbval 32103 | The value of the operator resulting from the outer product ∣ 𝐴⟩ ⟨𝐵 ∣ of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴)) | ||
| Theorem | kbmul 32104 | Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) ·ℎ 𝐶))) | ||
| Theorem | kbpj 32105 | If a vector 𝐴 has norm 1, the outer product ∣ 𝐴⟩⟨𝐴 ∣ is the projector onto the subspace spanned by 𝐴. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴}))) | ||
| Theorem | eleigvec 32106* | Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) | ||
| Theorem | eleigvec2 32107 | Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ ℋ → (𝐴 ∈ (eigvec‘𝑇) ↔ (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ∧ (𝑇‘𝐴) ∈ (span‘{𝐴})))) | ||
| Theorem | eleigveccl 32108 | Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → 𝐴 ∈ ℋ) | ||
| Theorem | eigvalval 32109 | The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) = (((𝑇‘𝐴) ·ih 𝐴) / ((normℎ‘𝐴)↑2))) | ||
| Theorem | eigvalcl 32110 | An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℂ) | ||
| Theorem | eigvec1 32111 | Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇‘𝐴) = (((eigval‘𝑇)‘𝐴) ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) | ||
| Theorem | eighmre 32112 | The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((eigval‘𝑇)‘𝐴) ∈ ℝ) | ||
| Theorem | eighmorth 32113 | Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ HrmOp ∧ 𝐴 ∈ (eigvec‘𝑇)) ∧ (𝐵 ∈ (eigvec‘𝑇) ∧ ((eigval‘𝑇)‘𝐴) ≠ ((eigval‘𝑇)‘𝐵))) → (𝐴 ·ih 𝐵) = 0) | ||
| Theorem | nmopnegi 32114 | Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 32180, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (normop‘(-1 ·op 𝑇)) = (normop‘𝑇) | ||
| Theorem | lnop0 32115 | The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇‘0ℎ) = 0ℎ) | ||
| Theorem | lnopmul 32116 | Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopli 32117 | Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))) | ||
| Theorem | lnopfi 32118 | A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ 𝑇: ℋ⟶ ℋ | ||
| Theorem | lnop0i 32119 | The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇‘0ℎ) = 0ℎ | ||
| Theorem | lnopaddi 32120 | Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopmuli 32121 | Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopaddmuli 32122 | Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) | ||
| Theorem | lnopsubi 32123 | Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −ℎ 𝐵)) = ((𝑇‘𝐴) −ℎ (𝑇‘𝐵))) | ||
| Theorem | lnopsubmuli 32124 | Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) −ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) | ||
| Theorem | lnopmulsubi 32125 | Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) −ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) −ℎ (𝑇‘𝐶))) | ||
| Theorem | homco2 32126 | Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 31950 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ (𝐴 ·op 𝑈)) = (𝐴 ·op (𝑇 ∘ 𝑈))) | ||
| Theorem | idunop 32127 | The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ UniOp | ||
| Theorem | 0cnop 32128 | The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ ContOp | ||
| Theorem | 0cnfn 32129 | The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( ℋ × {0}) ∈ ContFn | ||
| Theorem | idcnop 32130 | The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ ContOp | ||
| Theorem | idhmop 32131 | The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.) |
| ⊢ Iop ∈ HrmOp | ||
| Theorem | 0hmop 32132 | The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ HrmOp | ||
| Theorem | 0lnop 32133 | The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ LinOp | ||
| Theorem | 0lnfn 32134 | The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ( ℋ × {0}) ∈ LinFn | ||
| Theorem | nmop0 32135 | The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) |
| ⊢ (normop‘ 0hop ) = 0 | ||
| Theorem | nmfn0 32136 | The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (normfn‘( ℋ × {0})) = 0 | ||
| Theorem | hmopbdoptHIL 32137 | A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ BndLinOp) | ||
| Theorem | hoddii 32138 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31929 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑅 ∈ LinOp & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇)) | ||
| Theorem | hoddi 32139 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31929 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑅 ∈ LinOp ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑅 ∘ (𝑆 −op 𝑇)) = ((𝑅 ∘ 𝑆) −op (𝑅 ∘ 𝑇))) | ||
| Theorem | nmop0h 32140 | The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 32163.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
| ⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0) | ||
| Theorem | idlnop 32141 | The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ ( I ↾ ℋ) ∈ LinOp | ||
| Theorem | 0bdop 32142 | The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ BndLinOp | ||
| Theorem | adj0 32143 | Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ (adjℎ‘ 0hop ) = 0hop | ||
| Theorem | nmlnop0iALT 32144 | A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | nmlnop0iHIL 32145 | A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | nmlnopgt0i 32146 | A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇 ≠ 0hop ↔ 0 < (normop‘𝑇)) | ||
| Theorem | nmlnop0 32147 | A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) = 0 ↔ 𝑇 = 0hop )) | ||
| Theorem | nmlnopne0 32148 | A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → ((normop‘𝑇) ≠ 0 ↔ 𝑇 ≠ 0hop )) | ||
| Theorem | lnopmi 32149 | The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp) | ||
| Theorem | lnophsi 32150 | The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 +op 𝑇) ∈ LinOp | ||
| Theorem | lnophdi 32151 | The difference of two linear operators is linear. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 −op 𝑇) ∈ LinOp | ||
| Theorem | lnopcoi 32152 | The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑆 ∘ 𝑇) ∈ LinOp | ||
| Theorem | lnopco0i 32153 | The composition of a linear operator with one whose norm is zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ LinOp & ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((normop‘𝑇) = 0 → (normop‘(𝑆 ∘ 𝑇)) = 0) | ||
| Theorem | lnopeq0lem1 32154 | Lemma for lnopeq0i 32156. Apply the generalized polarization identity polid2i 31306 to the quadratic form ((𝑇‘𝑥), 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4) | ||
| Theorem | lnopeq0lem2 32155 | Lemma for lnopeq0i 32156. (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (((((𝑇‘(𝐴 +ℎ 𝐵)) ·ih (𝐴 +ℎ 𝐵)) − ((𝑇‘(𝐴 −ℎ 𝐵)) ·ih (𝐴 −ℎ 𝐵))) + (i · (((𝑇‘(𝐴 +ℎ (i ·ℎ 𝐵))) ·ih (𝐴 +ℎ (i ·ℎ 𝐵))) − ((𝑇‘(𝐴 −ℎ (i ·ℎ 𝐵))) ·ih (𝐴 −ℎ (i ·ℎ 𝐵)))))) / 4)) | ||
| Theorem | lnopeq0i 32156* | A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 31977 for arbitrary operators, when the operator is linear we need to consider only the values of the quadratic form (𝑇‘𝑥) ·ih 𝑥). (Contributed by NM, 26-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = 0 ↔ 𝑇 = 0hop ) | ||
| Theorem | lnopeqi 32157* | Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑈 ∈ LinOp ⇒ ⊢ (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈) | ||
| Theorem | lnopeq 32158* | Two linear Hilbert space operators are equal iff their quadratic forms are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑈 ∈ LinOp) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) = ((𝑈‘𝑥) ·ih 𝑥) ↔ 𝑇 = 𝑈)) | ||
| Theorem | lnopunilem1 32159* | Lemma for lnopunii 32161. (Contributed by NM, 14-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℂ ⇒ ⊢ (ℜ‘(𝐶 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝐶 · (𝐴 ·ih 𝐵))) | ||
| Theorem | lnopunilem2 32160* | Lemma for lnopunii 32161. (Contributed by NM, 12-May-2005.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) & ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) | ||
| Theorem | lnopunii 32161* | If a linear operator (whose range is ℋ) is idempotent in the norm, the operator is unitary. Similar to theorem in [AkhiezerGlazman] p. 73. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇: ℋ–onto→ ℋ & ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) ⇒ ⊢ 𝑇 ∈ UniOp | ||
| Theorem | elunop2 32162* | An operator is unitary iff it is linear, onto, and idempotent in the norm. Similar to theorem in [AkhiezerGlazman] p. 73, and its converse. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp ↔ (𝑇 ∈ LinOp ∧ 𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥))) | ||
| Theorem | nmopun 32163 | Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
| ⊢ (( ℋ ≠ 0ℋ ∧ 𝑇 ∈ UniOp) → (normop‘𝑇) = 1) | ||
| Theorem | unopbd 32164 | A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp) | ||
| Theorem | lnophmlem1 32165* | Lemma for lnophmi 32167. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ⇒ ⊢ (𝐴 ·ih (𝑇‘𝐴)) ∈ ℝ | ||
| Theorem | lnophmlem2 32166* | Lemma for lnophmi 32167. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ⇒ ⊢ (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) | ||
| Theorem | lnophmi 32167* | A linear operator is Hermitian if 𝑥 ·ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ ⇒ ⊢ 𝑇 ∈ HrmOp | ||
| Theorem | lnophm 32168* | A linear operator is Hermitian if 𝑥 ·ih (𝑇‘𝑥) takes only real values. Remark in [ReedSimon] p. 195. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ ∀𝑥 ∈ ℋ (𝑥 ·ih (𝑇‘𝑥)) ∈ ℝ) → 𝑇 ∈ HrmOp) | ||
| Theorem | hmops 32169 | The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 +op 𝑈) ∈ HrmOp) | ||
| Theorem | hmopm 32170 | The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) → (𝐴 ·op 𝑇) ∈ HrmOp) | ||
| Theorem | hmopd 32171 | The difference of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 −op 𝑈) ∈ HrmOp) | ||
| Theorem | hmopco 32172 | The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp) | ||
| Theorem | nmbdoplbi 32173 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmbdoplb 32174 | A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ BndLinOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcexi 32175* | Lemma for nmcopexi 32176 and nmcfnexi 32200. The norm of a continuous linear Hilbert space operator or functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by Mario Carneiro, 17-Nov-2013.) (Proof shortened by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) & ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < ) & ⊢ (𝑥 ∈ ℋ → (𝑁‘(𝑇‘𝑥)) ∈ ℝ) & ⊢ (𝑁‘(𝑇‘0ℎ)) = 0 & ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) ⇒ ⊢ (𝑆‘𝑇) ∈ ℝ | ||
| Theorem | nmcopexi 32176 | The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ (normop‘𝑇) ∈ ℝ | ||
| Theorem | nmcoplbi 32177 | A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcopex 32178 | The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (normop‘𝑇) ∈ ℝ) | ||
| Theorem | nmcoplb 32179 | A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmophmi 32180 | The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇))) | ||
| Theorem | bdophmi 32181 | The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ BndLinOp) | ||
| Theorem | lnconi 32182* | Lemma for lnopconi 32183 and lnfnconi 32204. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ 𝐶 → 𝑆 ∈ ℝ) & ⊢ ((𝑇 ∈ 𝐶 ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))) & ⊢ (𝑇 ∈ 𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧)) & ⊢ (𝑦 ∈ ℋ → (𝑁‘(𝑇‘𝑦)) ∈ ℝ) & ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥))) ⇒ ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
| Theorem | lnopconi 32183* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
| Theorem | lnopcon 32184* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) | ||
| Theorem | lnopcnbd 32185 | A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)) | ||
| Theorem | lncnopbd 32186 | A continuous linear operator is a bounded linear operator. This theorem justifies our use of "bounded linear" as an interchangeable condition for "continuous linear" used in some textbook proofs. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp) | ||
| Theorem | lncnbd 32187 | A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (LinOp ∩ ContOp) = BndLinOp | ||
| Theorem | lnopcnre 32188 | A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ (normop‘𝑇) ∈ ℝ)) | ||
| Theorem | lnfnli 32189 | Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))) | ||
| Theorem | lnfnfi 32190 | A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ 𝑇: ℋ⟶ℂ | ||
| Theorem | lnfn0i 32191 | The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ (𝑇‘0ℎ) = 0 | ||
| Theorem | lnfnaddi 32192 | Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) | ||
| Theorem | lnfnmuli 32193 | Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵))) | ||
| Theorem | lnfnaddmuli 32194 | Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶)))) | ||
| Theorem | lnfnsubi 32195 | Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵))) | ||
| Theorem | lnfn0 32196 | The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn → (𝑇‘0ℎ) = 0) | ||
| Theorem | lnfnmul 32197 | Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵))) | ||
| Theorem | nmbdfnlbi 32198 | A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) ⇒ ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmbdfnlb 32199 | A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcfnexi 32200 | The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ (normfn‘𝑇) ∈ ℝ | ||
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