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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hmopd 29801 | 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 29802 | 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 29803 | 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 29804 | 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 29805* | Lemma for nmcopexi 29806 and nmcfnexi 29830. 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 29806 | 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 29807 | 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 29808 | 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 29809 | 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 29810 | 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 29811 | 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 29812* | Lemma for lnopconi 29813 and lnfnconi 29834. (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 29813* | 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 29814* | 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 29815 | 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 29816 | 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 29817 | A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
⊢ (LinOp ∩ ContOp) = BndLinOp | ||
Theorem | lnopcnre 29818 | 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 29819 | 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 29820 | 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 29821 | 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 29822 | 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 29823 | 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 29824 | 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 29825 | 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 29826 | 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 29827 | 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 29828 | 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 29829 | 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 29830 | 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 29831 | 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 29832 | 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 29833 | 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 29834* | 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 29835* | 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 29836 | 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 29837 | 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 29838 | 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 29839 | 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 29840 | 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 29841* | 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 29842* | 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 29843* | A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 29845 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 29844* | 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 29845. For the continuous linear functional version, see riesz3i 29841 and riesz4 29843. (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 29845* | 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 29844. (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 29846* | Lemma for cnlnadji 29855 (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 29847* | Lemma for cnlnadji 29855. 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 29848* | Lemma for cnlnadji 29855. By riesz4 29843, 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 29849* | Lemma for cnlnadji 29855. 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 29850* | Lemma for cnlnadji 29855. 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 29851* | Lemma for cnlnadji 29855. 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 29852* | Lemma for cnlnadji 29855. 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 29853* | Lemma for cnlnadji 29855. 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 29854* | Lemma for cnlnadji 29855. 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 29855* | 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 29856* | 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 29857* | 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 29858* | 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 29859 | 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 29860 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ BndLinOp ⊆ dom adjℎ | ||
Theorem | bdopadj 29861 | Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom adjℎ) | ||
Theorem | adjbdln 29862 | The adjoint of a bounded linear operator is a bounded linear operator. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → (adjℎ‘𝑇) ∈ BndLinOp) | ||
Theorem | adjbdlnb 29863 | An operator is bounded and linear iff its adjoint is. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp ↔ (adjℎ‘𝑇) ∈ BndLinOp) | ||
Theorem | adjbd1o 29864 | The mapping of adjoints of bounded linear operators is one-to-one onto. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (adjℎ ↾ BndLinOp):BndLinOp–1-1-onto→BndLinOp | ||
Theorem | adjlnop 29865 | The adjoint of an operator is linear. Proposition 1 of [AkhiezerGlazman] p. 80. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ LinOp) | ||
Theorem | adjsslnop 29866 | Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
⊢ dom adjℎ ⊆ LinOp | ||
Theorem | nmopadjlei 29867 | Property of the norm of an adjoint. Part of proof of Theorem 3.10 of [Beran] p. 104. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝐴 ∈ ℋ → (normℎ‘((adjℎ‘𝑇)‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))) | ||
Theorem | nmopadjlem 29868 | Lemma for nmopadji 29869. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇) | ||
Theorem | nmopadji 29869 | Property of the norm of an adjoint. Theorem 3.11(v) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(adjℎ‘𝑇)) = (normop‘𝑇) | ||
Theorem | adjeq0 29870 | An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 = 0hop ↔ (adjℎ‘𝑇) = 0hop ) | ||
Theorem | adjmul 29871 | The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝐴 ·op 𝑇)) = ((∗‘𝐴) ·op (adjℎ‘𝑇))) | ||
Theorem | adjadd 29872 | The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇))) | ||
Theorem | nmoptrii 29873 | Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) | ||
Theorem | nmopcoi 29874 | Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑆 ∘ 𝑇)) ≤ ((normop‘𝑆) · (normop‘𝑇)) | ||
Theorem | bdophsi 29875 | The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 +op 𝑇) ∈ BndLinOp | ||
Theorem | bdophdi 29876 | The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 −op 𝑇) ∈ BndLinOp | ||
Theorem | bdopcoi 29877 | The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (𝑆 ∘ 𝑇) ∈ BndLinOp | ||
Theorem | nmoptri2i 29878 | Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ ((normop‘𝑆) − (normop‘𝑇)) ≤ (normop‘(𝑆 +op 𝑇)) | ||
Theorem | adjcoi 29879 | The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑆 ∈ BndLinOp & ⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (adjℎ‘(𝑆 ∘ 𝑇)) = ((adjℎ‘𝑇) ∘ (adjℎ‘𝑆)) | ||
Theorem | nmopcoadji 29880 | The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj2i 29881 | The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ (normop‘(𝑇 ∘ (adjℎ‘𝑇))) = ((normop‘𝑇)↑2) | ||
Theorem | nmopcoadj0i 29882 | An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑇 ∈ BndLinOp ⇒ ⊢ ((𝑇 ∘ (adjℎ‘𝑇)) = 0hop ↔ 𝑇 = 0hop ) | ||
Theorem | unierri 29883 | If we approximate a chain of unitary transformations (quantum computer gates) 𝐹, 𝐺 by other unitary transformations 𝑆, 𝑇, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝐹 ∈ UniOp & ⊢ 𝐺 ∈ UniOp & ⊢ 𝑆 ∈ UniOp & ⊢ 𝑇 ∈ UniOp ⇒ ⊢ (normop‘((𝐹 ∘ 𝐺) −op (𝑆 ∘ 𝑇))) ≤ ((normop‘(𝐹 −op 𝑆)) + (normop‘(𝐺 −op 𝑇))) | ||
Theorem | branmfn 29884 | The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (normℎ‘𝐴)) | ||
Theorem | brabn 29885 | The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) ∈ ℝ) | ||
Theorem | rnbra 29886 | The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ ran bra = (LinFn ∩ ContFn) | ||
Theorem | bra11 29887 | The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ bra: ℋ–1-1-onto→(LinFn ∩ ContFn) | ||
Theorem | bracnln 29888 | A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn)) | ||
Theorem | cnvbraval 29889* | Value of the converse of the bra function. Based on the Riesz Lemma riesz4 29843, this very important theorem not only justifies the Dirac bra-ket notation, but allows us to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from ℋ to ℂ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) = (℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))) | ||
Theorem | cnvbracl 29890 | Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (◡bra‘𝑇) ∈ ℋ) | ||
Theorem | cnvbrabra 29891 | The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (◡bra‘(bra‘𝐴)) = 𝐴) | ||
Theorem | bracnvbra 29892 | The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → (bra‘(◡bra‘𝑇)) = 𝑇) | ||
Theorem | bracnlnval 29893* | The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ (LinFn ∩ ContFn) → 𝑇 = (bra‘(℩𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦)))) | ||
Theorem | cnvbramul 29894 | Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ContFn)) → (◡bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) ·ℎ (◡bra‘𝑇))) | ||
Theorem | kbass1 29895 | Dirac bra-ket associative law ( ∣ 𝐴〉 〈𝐵 ∣ ) ∣ 𝐶〉 = ∣ 𝐴〉(〈𝐵 ∣ 𝐶〉) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 〈𝐵 ∣ 𝐶〉 is a complex number, it is the first argument in the inner product ·ℎ that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) ·ℎ 𝐴)) | ||
Theorem | kbass2 29896 | Dirac bra-ket associative law (〈𝐴 ∣ 𝐵〉)〈𝐶 ∣ = 〈𝐴 ∣ ( ∣ 𝐵〉 〈𝐶 ∣ ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶)) = ((bra‘𝐴) ∘ (𝐵 ketbra 𝐶))) | ||
Theorem | kbass3 29897 | Dirac bra-ket associative law 〈𝐴 ∣ 𝐵〉 〈𝐶 ∣ 𝐷〉 = (〈𝐴 ∣ 𝐵〉 〈𝐶 ∣ ) ∣ 𝐷〉. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷)) | ||
Theorem | kbass4 29898 | Dirac bra-ket associative law 〈𝐴 ∣ 𝐵〉 〈𝐶 ∣ 𝐷〉 = 〈𝐴 ∣ ( ∣ 𝐵〉 〈𝐶 ∣ 𝐷〉). (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) ·ℎ 𝐵))) | ||
Theorem | kbass5 29899 | Dirac bra-ket associative law ( ∣ 𝐴〉 〈𝐵 ∣ )( ∣ 𝐶〉 〈𝐷 ∣ ) = (( ∣ 𝐴〉 〈𝐵 ∣ ) ∣ 𝐶〉)〈𝐷 ∣. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)) | ||
Theorem | kbass6 29900 | Dirac bra-ket associative law ( ∣ 𝐴〉 〈𝐵 ∣ )( ∣ 𝐶〉 〈𝐷 ∣ ) = ∣ 𝐴〉 (〈𝐵 ∣ ( ∣ 𝐶〉 〈𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))))) |
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