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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeigvecval 29601* 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 → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})
 
Theoremeigvalfval 29602* The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))
 
Theoremspecval 29603* The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})
 
Theoremspeccl 29604 The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)
 
Theoremhhlnoi 29605 The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐿 = (𝑈 LnOp 𝑈)       LinOp = 𝐿
 
Theoremhhnmoi 29606 The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑁 = (𝑈 normOpOLD 𝑈)       normop = 𝑁
 
Theoremhhbloi 29607 A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐵 = (𝑈 BLnOp 𝑈)       BndLinOp = 𝐵
 
Theoremhh0oi 29608 The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑍 = (𝑈 0op 𝑈)        0hop = 𝑍
 
Theoremhhcno 29609 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐷 = (norm ∘ − )    &   𝐽 = (MetOpen‘𝐷)       ContOp = (𝐽 Cn 𝐽)
 
Theoremhhcnf 29610 The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐷 = (norm ∘ − )    &   𝐽 = (MetOpen‘𝐷)    &   𝐾 = (TopOpen‘ℂfld)       ContFn = (𝐽 Cn 𝐾)
 
Theoremdmadjrnb 29611 The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 6694.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj ↔ (adj𝑇) ∈ dom adj)
 
Theoremnmoplb 29612 A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (norm‘(𝑇𝐴)) ≤ (normop𝑇))
 
Theoremnmopub 29613* An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
 
Theoremnmopub2tALT 29614* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normop𝑇) ≤ 𝐴)
 
Theoremnmopub2tHIL 29615* An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normop𝑇) ≤ 𝐴)
 
Theoremnmopge0 29616 The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 0 ≤ (normop𝑇))
 
Theoremnmopgt0 29617 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: ℋ⟶ ℋ → ((normop𝑇) ≠ 0 ↔ 0 < (normop𝑇)))
 
Theoremcnopc 29618* Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
 
Theoremlnopl 29619 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
 
Theoremunop 29620 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))
 
Theoremunopf1o 29621 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
 
Theoremunopnorm 29622 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (norm‘(𝑇𝐴)) = (norm𝐴))
 
Theoremcnvunop 29623 The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ UniOp)
 
Theoremunopadj 29624 The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (𝐴 ·ih (𝑇𝐵)))
 
Theoremunoplin 29625 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
 
Theoremcounop 29626 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆𝑇) ∈ UniOp)
 
Theoremhmop 29627 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵))
 
Theoremhmopre 29628 The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐴) ∈ ℝ)
 
Theoremnmfnlb 29629 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))
 
Theoremnmfnleub 29630* 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*) → ((normfn𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
 
Theoremnmfnleub2 29631* 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‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normfn𝑇) ≤ 𝐴)
 
Theoremnmfnge0 29632 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → 0 ≤ (normfn𝑇))
 
Theoremelnlfn 29633 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 ∧ (𝑇𝐴) = 0)))
 
Theoremelnlfn2 29634 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)) → (𝑇𝐴) = 0)
 
Theoremcnfnc 29635* 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 ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))
 
Theoremlnfnl 29636 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 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
 
Theoremadjcl 29637 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ) → ((adj𝑇)‘𝐴) ∈ ℋ)
 
Theoremadj1 29638 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))
 
Theoremadj2 29639 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (𝐴 ·ih ((adj𝑇)‘𝐵)))
 
Theoremadjeq 29640* 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 ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆𝑦))) → (adj𝑇) = 𝑆)
 
Theoremadjadj 29641 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj‘(adj𝑇)) = 𝑇)
 
Theoremadjvalval 29642* 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 adj𝐴 ∈ ℋ) → ((adj𝑇)‘𝐴) = (𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤)))
 
Theoremunopadj2 29643 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 → (adj𝑇) = 𝑇)
 
Theoremhmopadj 29644 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → (adj𝑇) = 𝑇)
 
Theoremhmdmadj 29645 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ dom adj)
 
Theoremhmopadj2 29646 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 adj → (𝑇 ∈ HrmOp ↔ (adj𝑇) = 𝑇))
 
Theoremhmoplin 29647 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
 
Theorembrafval 29648* The bra of a vector, expressed as 𝐴 in Dirac notation. See df-bra 29555. (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)))
 
Theorembraval 29649 A bra-ket juxtaposition, expressed as 𝐴𝐵 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))
 
Theorembraadd 29650 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 + 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)))
 
Theorembramul 29651 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 · 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))
 
Theorembrafn 29652 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)
 
Theorembralnfn 29653 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)
 
Theorembracl 29654 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ)
 
Theorembra0 29655 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‘0) = ( ℋ × {0})
 
Theorembrafnmul 29656 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 · 𝐵)) = ((∗‘𝐴) ·fn (bra‘𝐵)))
 
Theoremkbfval 29657* The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 29556. (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) · 𝐴)))
 
Theoremkbop 29658 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)
 
Theoremkbval 29659 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) · 𝐴))
 
Theoremkbmul 29660 Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) · 𝐶)))
 
Theoremkbpj 29661 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 ∧ (norm𝐴) = 1) → (𝐴 ketbra 𝐴) = (proj‘(span‘{𝐴})))
 
Theoremeleigvec 29662* 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 ≠ 0 ∧ ∃𝑥 ∈ ℂ (𝑇𝐴) = (𝑥 · 𝐴))))
 
Theoremeleigvec2 29663 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 ≠ 0 ∧ (𝑇𝐴) ∈ (span‘{𝐴}))))
 
Theoremeleigveccl 29664 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)
 
Theoremeigvalval 29665 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) = (((𝑇𝐴) ·ih 𝐴) / ((norm𝐴)↑2)))
 
Theoremeigvalcl 29666 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)
 
Theoremeigvec1 29667 Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ (eigvec‘𝑇)) → ((𝑇𝐴) = (((eigval‘𝑇)‘𝐴) · 𝐴) ∧ 𝐴 ≠ 0))
 
Theoremeighmre 29668 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)
 
Theoremeighmorth 29669 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)
 
Theoremnmopnegi 29670 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 29736, 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)) = (normop𝑇)
 
Theoremlnop0 29671 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‘0) = 0)
 
Theoremlnopmul 29672 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 · 𝐵)) = (𝐴 · (𝑇𝐵)))
 
Theoremlnopli 29673 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 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
 
Theoremlnopfi 29674 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
 
Theoremlnop0i 29675 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‘0) = 0
 
Theoremlnopaddi 29676 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 + 𝐵)) = ((𝑇𝐴) + (𝑇𝐵)))
 
Theoremlnopmuli 29677 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 · 𝐵)) = (𝐴 · (𝑇𝐵)))
 
Theoremlnopaddmuli 29678 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 + (𝐴 · 𝐶))) = ((𝑇𝐵) + (𝐴 · (𝑇𝐶))))
 
Theoremlnopsubi 29679 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 𝐵)) = ((𝑇𝐴) − (𝑇𝐵)))
 
Theoremlnopsubmuli 29680 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 (𝐴 · 𝐶))) = ((𝑇𝐵) − (𝐴 · (𝑇𝐶))))
 
Theoremlnopmulsubi 29681 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 · 𝐵) − 𝐶)) = ((𝐴 · (𝑇𝐵)) − (𝑇𝐶)))
 
Theoremhomco2 29682 Move a scalar product out of a composition of operators. The operator 𝑇 must be linear, unlike homco1 29506 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 (𝑇𝑈)))
 
Theoremidunop 29683 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ UniOp
 
Theorem0cnop 29684 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ ContOp
 
Theorem0cnfn 29685 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ ContFn
 
Theoremidcnop 29686 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ ContOp
 
Theoremidhmop 29687 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)
Iop ∈ HrmOp
 
Theorem0hmop 29688 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
0hop ∈ HrmOp
 
Theorem0lnop 29689 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop ∈ LinOp
 
Theorem0lnfn 29690 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
( ℋ × {0}) ∈ LinFn
 
Theoremnmop0 29691 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)
(normop‘ 0hop ) = 0
 
Theoremnmfn0 29692 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(normfn‘( ℋ × {0})) = 0
 
TheoremhmopbdoptHIL 29693 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)
 
Theoremhoddii 29694 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 29485 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
𝑅 ∈ LinOp    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       (𝑅 ∘ (𝑆op 𝑇)) = ((𝑅𝑆) −op (𝑅𝑇))
 
Theoremhoddi 29695 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 29485 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
((𝑅 ∈ LinOp ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑅 ∘ (𝑆op 𝑇)) = ((𝑅𝑆) −op (𝑅𝑇)))
 
Theoremnmop0h 29696 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0 in nmopun 29719.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)
(( ℋ = 0𝑇: ℋ⟶ ℋ) → (normop𝑇) = 0)
 
Theoremidlnop 29697 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)
( I ↾ ℋ) ∈ LinOp
 
Theorem0bdop 29698 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
0hop ∈ BndLinOp
 
Theoremadj0 29699 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(adj‘ 0hop ) = 0hop
 
Theoremnmlnop0iALT 29700 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       ((normop𝑇) = 0 ↔ 𝑇 = 0hop )
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