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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmopre 27901 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 27902 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 27903* 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 27904* 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 27905 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 27906 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
HrmOp ∈ V
 
Theoremnmfnval 27907* 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 27908* The set in the supremum of the functional norm definition df-nmfn 27876 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 27909* The set in the supremum of the functional norm definition df-nmfn 27876 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(abs‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}
 
Theoremnmfnxr 27910 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 27911 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 27912 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 27913* 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 27914* 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 27915 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)
 
Theoremdfadj2 27916* 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 27917 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Fun adj
 
Theoremdmadjss 27918 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 27919 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 27920* 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 27921* 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 27922* 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 27923 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
adj = adj
 
Theoremfuncnvadj 27924 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)
Fun adj
 
Theoremadj1o 27925 The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
adj:dom adj1-1-onto→dom adj
 
Theoremdmadjrn 27926 The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
 
Theoremeigvecval 27927* 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 27928* 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 27929* 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 27930 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 27931 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 27932 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 27933 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 27934 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 27935 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 27936 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 27937 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 6012.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj ↔ (adj𝑇) ∈ dom adj)
 
Theoremnmoplb 27938 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 27939* 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 27940* 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 27941* 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 27942 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 27943 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 27944* 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 27945 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 27946 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 27947 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 27948 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 27949 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 27950 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 27951 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 27952 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 27953 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 27954 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 27955 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 27956* 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 27957* 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 27958 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 27959 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 27960 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 27961* 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 27962 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 27963 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 27964 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 27965 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 27966* 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 27967 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 27968* 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 27969 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 27970 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → (adj𝑇) = 𝑇)
 
Theoremhmdmadj 27971 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ dom adj)
 
Theoremhmopadj2 27972 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 27973 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇 ∈ LinOp)
 
Theorembrafval 27974* The bra of a vector, expressed as 𝐴 in Dirac notation. See df-bra 27881. (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 27975 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 27976 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 27977 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 27978 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 27979 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 27980 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) ∈ ℂ)
 
Theorembra0 27981 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 27982 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 27983* The outer product of two vectors, expressed as 𝐴 𝐵 in Dirac notation. See df-kb 27882. (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 27984 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 27985 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 27986 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 27987 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 27988* 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 27989 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 27990 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 27991 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 27992 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 27993 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 27994 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 27995 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 27996 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 28062, 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 27997 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 27998 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 27999 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 28000 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
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