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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nmfnlb 29701 | A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇)) | ||
Theorem | nmfnleub 29702* | An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
Theorem | nmfnleub2 29703* | An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴) | ||
Theorem | nmfnge0 29704 | The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) | ||
Theorem | elnlfn 29705 | Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇‘𝐴) = 0))) | ||
Theorem | elnlfn2 29706 | Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ (null‘𝑇)) → (𝑇‘𝐴) = 0) | ||
Theorem | cnfnc 29707* | Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)) | ||
Theorem | lnfnl 29708 | Basic linearity property of a linear 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 | adjcl 29709 | Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) | ||
Theorem | adj1 29710 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)) | ||
Theorem | adj2 29711 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) | ||
Theorem | adjeq 29712* | A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆‘𝑦))) → (adjℎ‘𝑇) = 𝑆) | ||
Theorem | adjadj 29713 | Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑇)) = 𝑇) | ||
Theorem | adjvalval 29714* | Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) | ||
Theorem | unopadj2 29715 | The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) | ||
Theorem | hmopadj 29716 | A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → (adjℎ‘𝑇) = 𝑇) | ||
Theorem | hmdmadj 29717 | Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) | ||
Theorem | hmopadj2 29718 | An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (𝑇 ∈ HrmOp ↔ (adjℎ‘𝑇) = 𝑇)) | ||
Theorem | hmoplin 29719 | A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | ||
Theorem | brafval 29720* | The bra of a vector, expressed as 〈𝐴 ∣ in Dirac notation. See df-bra 29627. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴))) | ||
Theorem | braval 29721 | A bra-ket juxtaposition, expressed as 〈𝐴 ∣ 𝐵〉 in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) | ||
Theorem | braadd 29722 | Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶))) | ||
Theorem | bramul 29723 | Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶))) | ||
Theorem | brafn 29724 | The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ) | ||
Theorem | bralnfn 29725 | The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ LinFn) | ||
Theorem | bracl 29726 | Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ) | ||
Theorem | bra0 29727 | The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ (bra‘0ℎ) = ( ℋ × {0}) | ||
Theorem | brafnmul 29728 | Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (bra‘(𝐴 ·ℎ 𝐵)) = ((∗‘𝐴) ·fn (bra‘𝐵))) | ||
Theorem | kbfval 29729* | The outer product of two vectors, expressed as ∣ 𝐴〉 〈𝐵 ∣ in Dirac notation. See df-kb 29628. (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 29730 | 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 29731 | 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 29732 | 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 29733 | 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 29734* | 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 29735 | 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 29736 | 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 29737 | 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 29738 | 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 29739 | 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 29740 | 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 29741 | 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 29742 | Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 29808, 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 29743 | 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 29744 | 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 29745 | 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 29746 | 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 29747 | 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 29748 | 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 29749 | 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 29750 | 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 29751 | 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 29752 | 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 29753 | 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 29754 | Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 29578 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 29755 | 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 29756 | The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
⊢ 0hop ∈ ContOp | ||
Theorem | 0cnfn 29757 | 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 29758 | 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 29759 | The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.) |
⊢ Iop ∈ HrmOp | ||
Theorem | 0hmop 29760 | The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
⊢ 0hop ∈ HrmOp | ||
Theorem | 0lnop 29761 | The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
⊢ 0hop ∈ LinOp | ||
Theorem | 0lnfn 29762 | 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 29763 | The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) |
⊢ (normop‘ 0hop ) = 0 | ||
Theorem | nmfn0 29764 | 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 29765 | 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 29766 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 29557 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 29767 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 29557 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 29768 | The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 29791.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0) | ||
Theorem | idlnop 29769 | 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 29770 | The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ 0hop ∈ BndLinOp | ||
Theorem | adj0 29771 | Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ (adjℎ‘ 0hop ) = 0hop | ||
Theorem | nmlnop0iALT 29772 | 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 29773 | 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 29774 | 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 29775 | 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 29776 | 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 29777 | 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 29778 | 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 29779 | 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 29780 | 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 29781 | 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 29782 | Lemma for lnopeq0i 29784. Apply the generalized polarization identity polid2i 28934 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 29783 | Lemma for lnopeq0i 29784. (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 29784* | A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 29605 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 29785* | 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 29786* | 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 29787* | Lemma for lnopunii 29789. (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 29788* | Lemma for lnopunii 29789. (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 29789* | 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 29790* | 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 29791 | 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 29792 | A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp) | ||
Theorem | lnophmlem1 29793* | Lemma for lnophmi 29795. (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 29794* | Lemma for lnophmi 29795. (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 29795* | 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 29796* | 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 29797 | 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 29798 | 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 29799 | 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 29800 | 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) |
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