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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremellnop 28101* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnopf 28102 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)

Theoremelbdop 28103 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))

Theorembdopln 28104 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Theorembdopf 28105 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)

TheoremnmopsetretALT 28106* The set in the supremum of the operator norm definition df-nmop 28082 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

TheoremnmopsetretHIL 28107* The set in the supremum of the operator norm definition df-nmop 28082 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmopsetn0 28108* The set in the supremum of the operator norm definition df-nmop 28082 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(norm‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}

Theoremnmopxr 28109 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (normop𝑇) ∈ ℝ*)

Theoremnmoprepnf 28110 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) ≠ +∞))

Theoremnmopgtmnf 28111 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → -∞ < (normop𝑇))

Theoremnmopreltpnf 28112 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) < +∞))

Theoremnmopre 28113 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)

Theoremelbdop2 28114 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) ∈ ℝ))

Theoremelunop 28115* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))

Theoremelhmop 28116* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))

Theoremhmopf 28117 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)

Theoremhmopex 28118 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
HrmOp ∈ V

Theoremnmfnval 28119* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))

Theoremnmfnsetre 28120* The set in the supremum of the functional norm definition df-nmfn 28088 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmfnsetn0 28121* The set in the supremum of the functional norm definition df-nmfn 28088 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(abs‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}

Theoremnmfnxr 28122 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) ∈ ℝ*)

Theoremnmfnrepnf 28123 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → ((normfn𝑇) ∈ ℝ ↔ (normfn𝑇) ≠ +∞))

Theoremnlfnval 28124 Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))

Theoremelcnfn 28125* Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ ConFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))

Theoremellnfn 28126* Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnfnf 28127 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Theoremdfadj2 28128* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}

Theoremfunadj 28129 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjss 28130 The domain of the adjoint function is a subset of the maps from to . (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
dom adj ⊆ ( ℋ ↑𝑚 ℋ)

Theoremdmadjop 28131 A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)

Theoremadjeu 28132* Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj ↔ ∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))

Theoremadjval 28133* Value of the adjoint function for 𝑇 in the domain of adj. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))

Theoremadjval2 28134* Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢𝑦))))

Theoremcnvadj 28135 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremfuncnvadj 28136 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)

Theoremadj1o 28137 The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjrn 28138 The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremeigvecval 28139* 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 28140* 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 28141* 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 28142 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 28143 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 28144 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 28145 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 28146 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 28147 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐷 = (norm ∘ − )    &   𝐽 = (MetOpen‘𝐷)       ConOp = (𝐽 Cn 𝐽)

Theoremhhcnf 28148 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)       ConFn = (𝐽 Cn 𝐾)

Theoremdmadjrnb 28149 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 6128.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremnmoplb 28150 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 28151* 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 28152* 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 28153* 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 28154 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 28155 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 28156* 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 ∈ ConOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))

Theoremlnopl 28157 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 28158 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 28159 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 28160 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 28161 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 28162 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 28163 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 28164 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 28165 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 28166 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 28167 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 28168* 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 28169* 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 28170 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 28171 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 28172 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 28173* 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 ∈ ConFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (abs‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))

Theoremlnfnl 28174 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 28175 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremadj1 28176 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 28177 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 28178* 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 28179 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjvalval 28180* 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 28181 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 28182 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → (adj𝑇) = 𝑇)

Theoremhmdmadj 28183 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ dom adj)

Theoremhmopadj2 28184 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 28185 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)

Theorembrafval 28186* The bra of a vector, expressed as 𝐴 in Dirac notation. See df-bra 28093. (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 28187 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 28188 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 28189 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 28190 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 28191 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 28192 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ)

Theorembra0 28193 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 28194 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 28195* The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 28094. (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 28196 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 28197 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 28198 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 28199 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 28200* 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))))

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