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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | idhmop 32001 | The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.) |
| ⊢ Iop ∈ HrmOp | ||
| Theorem | 0hmop 32002 | The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ HrmOp | ||
| Theorem | 0lnop 32003 | The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ LinOp | ||
| Theorem | 0lnfn 32004 | 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 32005 | The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.) |
| ⊢ (normop‘ 0hop ) = 0 | ||
| Theorem | nmfn0 32006 | 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 32007 | 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 32008 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31799 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 32009 | Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 31799 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 32010 | The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need ℋ ≠ 0ℋ in nmopun 32033.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.) |
| ⊢ (( ℋ = 0ℋ ∧ 𝑇: ℋ⟶ ℋ) → (normop‘𝑇) = 0) | ||
| Theorem | idlnop 32011 | 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 32012 | The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 0hop ∈ BndLinOp | ||
| Theorem | adj0 32013 | Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ (adjℎ‘ 0hop ) = 0hop | ||
| Theorem | nmlnop0iALT 32014 | 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 32015 | 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 32016 | 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 32017 | 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 32018 | 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 32019 | 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 32020 | 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 32021 | 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 32022 | 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 32023 | 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 32024 | Lemma for lnopeq0i 32026. Apply the generalized polarization identity polid2i 31176 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 32025 | Lemma for lnopeq0i 32026. (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 32026* | A condition implying that a linear Hilbert space operator is identically zero. Unlike ho01i 31847 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 32027* | 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 32028* | 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 32029* | Lemma for lnopunii 32031. (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 32030* | Lemma for lnopunii 32031. (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 32031* | 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 32032* | 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 32033 | 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 32034 | A unitary operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ BndLinOp) | ||
| Theorem | lnophmlem1 32035* | Lemma for lnophmi 32037. (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 32036* | Lemma for lnophmi 32037. (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 32037* | 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 32038* | 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 32039 | 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 32040 | 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 32041 | 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 32042 | 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) | ||
| Theorem | nmbdoplbi 32043 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmbdoplb 32044 | A lower bound for the norm of a bounded linear Hilbert space operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ BndLinOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcexi 32045* | Lemma for nmcopexi 32046 and nmcfnexi 32070. The norm of a continuous linear Hilbert space operator or functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by Mario Carneiro, 17-Nov-2013.) (Proof shortened by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) & ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < ) & ⊢ (𝑥 ∈ ℋ → (𝑁‘(𝑇‘𝑥)) ∈ ℝ) & ⊢ (𝑁‘(𝑇‘0ℎ)) = 0 & ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥)))) ⇒ ⊢ (𝑆‘𝑇) ∈ ℝ | ||
| Theorem | nmcopexi 32046 | The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ (normop‘𝑇) ∈ ℝ | ||
| Theorem | nmcoplbi 32047 | A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcopex 32048 | The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp) → (normop‘𝑇) ∈ ℝ) | ||
| Theorem | nmcoplb 32049 | A lower bound for the norm of a continuous linear Hilbert space operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinOp ∧ 𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmophmi 32050 | The norm of the scalar product of a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℂ → (normop‘(𝐴 ·op 𝑇)) = ((abs‘𝐴) · (normop‘𝑇))) | ||
| Theorem | bdophmi 32051 | The scalar product of a bounded linear operator is a bounded linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ BndLinOp) | ||
| Theorem | lnconi 32052* | Lemma for lnopconi 32053 and lnfnconi 32074. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ 𝐶 → 𝑆 ∈ ℝ) & ⊢ ((𝑇 ∈ 𝐶 ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))) & ⊢ (𝑇 ∈ 𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧)) & ⊢ (𝑦 ∈ ℋ → (𝑁‘(𝑇‘𝑦)) ∈ ℝ) & ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥))) ⇒ ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
| Theorem | lnopconi 32053* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
| Theorem | lnopcon 32054* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (normℎ‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) | ||
| Theorem | lnopcnbd 32055 | A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ 𝑇 ∈ BndLinOp)) | ||
| Theorem | lncnopbd 32056 | A continuous linear operator is a bounded linear operator. This theorem justifies our use of "bounded linear" as an interchangeable condition for "continuous linear" used in some textbook proofs. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) ↔ 𝑇 ∈ BndLinOp) | ||
| Theorem | lncnbd 32057 | A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (LinOp ∩ ContOp) = BndLinOp | ||
| Theorem | lnopcnre 32058 | A linear operator is continuous iff it is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinOp → (𝑇 ∈ ContOp ↔ (normop‘𝑇) ∈ ℝ)) | ||
| Theorem | lnfnli 32059 | Basic property of a linear Hilbert space 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 | lnfnfi 32060 | A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ 𝑇: ℋ⟶ℂ | ||
| Theorem | lnfn0i 32061 | The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ (𝑇‘0ℎ) = 0 | ||
| Theorem | lnfnaddi 32062 | Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) | ||
| Theorem | lnfnmuli 32063 | Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵))) | ||
| Theorem | lnfnaddmuli 32064 | Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶)))) | ||
| Theorem | lnfnsubi 32065 | Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵))) | ||
| Theorem | lnfn0 32066 | The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn → (𝑇‘0ℎ) = 0) | ||
| Theorem | lnfnmul 32067 | Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 · (𝑇‘𝐵))) | ||
| Theorem | nmbdfnlbi 32068 | A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) ⇒ ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmbdfnlb 32069 | A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcfnexi 32070 | The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ (normfn‘𝑇) ∈ ℝ | ||
| Theorem | nmcfnlbi 32071 | A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | nmcfnex 32072 | The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn) → (normfn‘𝑇) ∈ ℝ) | ||
| Theorem | nmcfnlb 32073 | A lower bound of the norm of a continuous linear Hilbert space functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ 𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | lnfnconi 32074* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) | ||
| Theorem | lnfncon 32075* | A condition equivalent to "𝑇 is continuous" when 𝑇 is linear. Theorem 3.5(iii) of [Beran] p. 99. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (abs‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))) | ||
| Theorem | lnfncnbd 32076 | A linear functional is continuous iff it is bounded. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn → (𝑇 ∈ ContFn ↔ (normfn‘𝑇) ∈ ℝ)) | ||
| Theorem | imaelshi 32077 | The image of a subspace under a linear operator is a subspace. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝐴 ∈ Sℋ ⇒ ⊢ (𝑇 “ 𝐴) ∈ Sℋ | ||
| Theorem | rnelshi 32078 | The range of a linear operator is a subspace. (Contributed by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp ⇒ ⊢ ran 𝑇 ∈ Sℋ | ||
| Theorem | nlelshi 32079 | The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn ⇒ ⊢ (null‘𝑇) ∈ Sℋ | ||
| Theorem | nlelchi 32080 | The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ (null‘𝑇) ∈ Cℋ | ||
| Theorem | riesz3i 32081* | A continuous linear functional can be expressed as an inner product. Existence part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ ∃𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤) | ||
| Theorem | riesz4i 32082* | A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinFn & ⊢ 𝑇 ∈ ContFn ⇒ ⊢ ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤) | ||
| Theorem | riesz4 32083* | A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 32085 for the bounded linear functional version. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinFn ∩ ContFn) → ∃!𝑤 ∈ ℋ ∀𝑣 ∈ ℋ (𝑇‘𝑣) = (𝑣 ·ih 𝑤)) | ||
| Theorem | riesz1 32084* | Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2 32085. For the continuous linear functional version, see riesz3i 32081 and riesz4 32083. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ LinFn → ((normfn‘𝑇) ∈ ℝ ↔ ∃𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) | ||
| Theorem | riesz2 32085* | Part 2 of the Riesz representation theorem for bounded linear functionals. The value of a bounded linear functional corresponds to a unique inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 1, see riesz1 32084. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)) | ||
| Theorem | cnlnadjlem1 32086* | Lemma for cnlnadji 32095 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional 𝐺. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) ⇒ ⊢ (𝐴 ∈ ℋ → (𝐺‘𝐴) = ((𝑇‘𝐴) ·ih 𝑦)) | ||
| Theorem | cnlnadjlem2 32087* | Lemma for cnlnadji 32095. 𝐺 is a continuous linear functional. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) ⇒ ⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈ ContFn)) | ||
| Theorem | cnlnadjlem3 32088* | Lemma for cnlnadji 32095. By riesz4 32083, 𝐵 is the unique vector such that (𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤) for all 𝑣. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) ⇒ ⊢ (𝑦 ∈ ℋ → 𝐵 ∈ ℋ) | ||
| Theorem | cnlnadjlem4 32089* | Lemma for cnlnadji 32095. The values of auxiliary function 𝐹 are vectors. (Contributed by NM, 17-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ ℋ → (𝐹‘𝐴) ∈ ℋ) | ||
| Theorem | cnlnadjlem5 32090* | Lemma for cnlnadji 32095. 𝐹 is an adjoint of 𝑇 (later, we will show it is unique). (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐶) ·ih 𝐴) = (𝐶 ·ih (𝐹‘𝐴))) | ||
| Theorem | cnlnadjlem6 32091* | Lemma for cnlnadji 32095. 𝐹 is linear. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ LinOp | ||
| Theorem | cnlnadjlem7 32092* | Lemma for cnlnadji 32095. Helper lemma to show that 𝐹 is continuous. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝐹‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
| Theorem | cnlnadjlem8 32093* | Lemma for cnlnadji 32095. 𝐹 is continuous. (Contributed by NM, 17-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ 𝐹 ∈ ContOp | ||
| Theorem | cnlnadjlem9 32094* | Lemma for cnlnadji 32095. 𝐹 provides an example showing the existence of a continuous linear adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp & ⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) & ⊢ 𝐵 = (℩𝑤 ∈ ℋ ∀𝑣 ∈ ℋ ((𝑇‘𝑣) ·ih 𝑦) = (𝑣 ·ih 𝑤)) & ⊢ 𝐹 = (𝑦 ∈ ℋ ↦ 𝐵) ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑧) = (𝑥 ·ih (𝑡‘𝑧)) | ||
| Theorem | cnlnadji 32095* | Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
| Theorem | cnlnadjeui 32096* | Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ 𝑇 ∈ LinOp & ⊢ 𝑇 ∈ ContOp ⇒ ⊢ ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦)) | ||
| Theorem | cnlnadjeu 32097* | Every continuous linear operator has a unique adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃!𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
| Theorem | cnlnadj 32098* | Every continuous linear operator has an adjoint. Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ (LinOp ∩ ContOp) → ∃𝑡 ∈ (LinOp ∩ ContOp)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑡‘𝑦))) | ||
| Theorem | cnlnssadj 32099 | Every continuous linear Hilbert space operator has an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
| ⊢ (LinOp ∩ ContOp) ⊆ dom adjℎ | ||
| Theorem | bdopssadj 32100 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ BndLinOp ⊆ dom adjℎ | ||
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