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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ellnop 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))) | ||
Theorem | lnopf 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) | ||
Theorem | elbdop 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 ∧ (normop‘𝑇) < +∞)) | ||
Theorem | bdopln 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) | ||
Theorem | bdopf 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) | ||
Theorem | nmopsetretALT 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 ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmopsetretHIL 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 ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmopsetn0 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.) |
⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} | ||
Theorem | nmopxr 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 → (normop‘𝑇) ∈ ℝ*) | ||
Theorem | nmoprepnf 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 → ((normop‘𝑇) ∈ ℝ ↔ (normop‘𝑇) ≠ +∞)) | ||
Theorem | nmopgtmnf 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 → -∞ < (normop‘𝑇)) | ||
Theorem | nmopreltpnf 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 → ((normop‘𝑇) ∈ ℝ ↔ (normop‘𝑇) < +∞)) | ||
Theorem | nmopre 28113 | The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | ||
Theorem | elbdop2 28114 | Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) ∈ ℝ)) | ||
Theorem | elunop 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 ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) | ||
Theorem | elhmop 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 (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | ||
Theorem | hmopf 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) | ||
Theorem | hmopex 28118 | The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
⊢ HrmOp ∈ V | ||
Theorem | nmfnval 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 → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) | ||
Theorem | nmfnsetre 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 ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmfnsetn0 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‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} | ||
Theorem | nmfnxr 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 → (normfn‘𝑇) ∈ ℝ*) | ||
Theorem | nmfnrepnf 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 → ((normfn‘𝑇) ∈ ℝ ↔ (normfn‘𝑇) ≠ +∞)) | ||
Theorem | nlfnval 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})) | ||
Theorem | elcnfn 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 ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑇‘𝑤) − (𝑇‘𝑥))) < 𝑦))) | ||
Theorem | ellnfn 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) | ||
Theorem | lnfnf 28127 | A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ) | ||
Theorem | dfadj2 28128* | Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ adjℎ = {〈𝑡, 𝑢〉 ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))} | ||
Theorem | funadj 28129 | Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ Fun adjℎ | ||
Theorem | dmadjss 28130 | The domain of the adjoint function is a subset of the maps from ℋ to ℋ. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ dom adjℎ ⊆ ( ℋ ↑𝑚 ℋ) | ||
Theorem | dmadjop 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 adjℎ → 𝑇: ℋ⟶ ℋ) | ||
Theorem | adjeu 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 adjℎ ↔ ∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))) | ||
Theorem | adjval 28133* | Value of the adjoint function for 𝑇 in the domain of adjℎ. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))) | ||
Theorem | adjval2 28134* | Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))) | ||
Theorem | cnvadj 28135 | The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ ◡adjℎ = adjℎ | ||
Theorem | funcnvadj 28136 | The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.) |
⊢ Fun ◡adjℎ | ||
Theorem | adj1o 28137 | The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | ||
Theorem | dmadjrn 28138 | The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) | ||
Theorem | eigvecval 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 ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)}) | ||
Theorem | eigvalfval 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) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | ||
Theorem | specval 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 ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | ||
Theorem | speccl 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) | ||
Theorem | hhlnoi 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐿 = (𝑈 LnOp 𝑈) ⇒ ⊢ LinOp = 𝐿 | ||
Theorem | hhnmoi 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) ⇒ ⊢ normop = 𝑁 | ||
Theorem | hhbloi 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) ⇒ ⊢ BndLinOp = 𝐵 | ||
Theorem | hh0oi 28146 | The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑍 = (𝑈 0op 𝑈) ⇒ ⊢ 0hop = 𝑍 | ||
Theorem | hhcno 28147 | The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ConOp = (𝐽 Cn 𝐽) | ||
Theorem | hhcnf 28148 | The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ ConFn = (𝐽 Cn 𝐾) | ||
Theorem | dmadjrnb 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.) |
⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) | ||
Theorem | nmoplb 28150 | A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇)) | ||
Theorem | nmopub 28151* | An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
Theorem | nmopub2tALT 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 (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) | ||
Theorem | nmopub2tHIL 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 (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) | ||
Theorem | nmopge0 28154 | The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) | ||
Theorem | nmopgt0 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 → ((normop‘𝑇) ≠ 0 ↔ 0 < (normop‘𝑇))) | ||
Theorem | cnopc 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 ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)) | ||
Theorem | lnopl 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 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))) | ||
Theorem | unop 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) → ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) | ||
Theorem | unopf1o 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) | ||
Theorem | unopnorm 28160 | A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘𝐴)) | ||
Theorem | cnvunop 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) | ||
Theorem | unopadj 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) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih (◡𝑇‘𝐵))) | ||
Theorem | unoplin 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) | ||
Theorem | counop 28164 | The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp) | ||
Theorem | hmop 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 (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)) | ||
Theorem | hmopre 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) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℝ) | ||
Theorem | nmfnlb 28167 | 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 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*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
Theorem | nmfnleub2 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‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴) | ||
Theorem | nmfnge0 28170 | 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 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 ∧ (𝑇‘𝐴) = 0))) | ||
Theorem | elnlfn2 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)) → (𝑇‘𝐴) = 0) | ||
Theorem | cnfnc 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 ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)) | ||
Theorem | lnfnl 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 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))) | ||
Theorem | adjcl 28175 | 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 28176 | 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 28177 | 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 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 ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆‘𝑦))) → (adjℎ‘𝑇) = 𝑆) | ||
Theorem | adjadj 28179 | 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 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 adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) | ||
Theorem | unopadj2 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 → (adjℎ‘𝑇) = ◡𝑇) | ||
Theorem | hmopadj 28182 | A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → (adjℎ‘𝑇) = 𝑇) | ||
Theorem | hmdmadj 28183 | Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) | ||
Theorem | hmopadj2 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 adjℎ → (𝑇 ∈ HrmOp ↔ (adjℎ‘𝑇) = 𝑇)) | ||
Theorem | hmoplin 28185 | A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | ||
Theorem | brafval 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))) | ||
Theorem | braval 28187 | 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 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 +ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶))) | ||
Theorem | bramul 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 ·ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶))) | ||
Theorem | brafn 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) | ||
Theorem | bralnfn 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) | ||
Theorem | bracl 28192 | 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 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‘0ℎ) = ( ℋ × {0}) | ||
Theorem | brafnmul 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 ·ℎ 𝐵)) = ((∗‘𝐴) ·fn (bra‘𝐵))) | ||
Theorem | kbfval 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) ·ℎ 𝐴))) | ||
Theorem | kbop 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) | ||
Theorem | kbval 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) ·ℎ 𝐴)) | ||
Theorem | kbmul 28198 | 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 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 ∧ (normℎ‘𝐴) = 1) → (𝐴 ketbra 𝐴) = (projℎ‘(span‘{𝐴}))) | ||
Theorem | eleigvec 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 ≠ 0ℎ ∧ ∃𝑥 ∈ ℂ (𝑇‘𝐴) = (𝑥 ·ℎ 𝐴)))) |
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