P. 296 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nmcfnex 29501	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 29502	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 29503*	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 29504*	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 29505	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 29506	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 29507	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 29508	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 29509	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
19.6.11  Riesz lemma
Theorem	riesz3i 29510*	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 29511*	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 29512*	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 29514 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 29513*	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 29514. For the continuous linear functional version, see riesz3i 29510 and riesz4 29512. (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 29514*	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 29513. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))
19.6.12  Adjoints (cont.)
Theorem	cnlnadjlem1 29515*	Lemma for cnlnadji 29524 (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 29516*	Lemma for cnlnadji 29524. 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 29517*	Lemma for cnlnadji 29524. By riesz4 29512, 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 29518*	Lemma for cnlnadji 29524. 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 29519*	Lemma for cnlnadji 29524. 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 29520*	Lemma for cnlnadji 29524. 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 29521*	Lemma for cnlnadji 29524. 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 29522*	Lemma for cnlnadji 29524. 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 29523*	Lemma for cnlnadji 29524. 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 29524*	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 29525*	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 29526*	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 29527*	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 29528	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 29529	Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
⊢ BndLinOp ⊆ dom adjℎ
Theorem	bdopadj 29530	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 29531	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 29532	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 29533	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 29534	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 29535	Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
⊢ dom adjℎ ⊆ LinOp
Theorem	nmopadjlei 29536	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 29537	Lemma for nmopadji 29538. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ BndLinOp    ⇒   ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇)
Theorem	nmopadji 29538	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 29539	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 29540	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 29541	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 29542	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 29543	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 29544	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 29545	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 29546	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 29547	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 29548	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 29549	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 29550	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 29551	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 )
19.6.13  Quantum computation error bound theorem
Theorem	unierri 29552	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)))
19.6.14  Dirac bra-ket notation (cont.)
Theorem	branmfn 29553	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 29554	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 29555	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 29556	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 29557	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 29558*	Value of the converse of the bra function. Based on the Riesz Lemma riesz4 29512, 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 29559	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 29560	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 29561	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 29562*	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 29563	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 29564	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 29565	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 29566	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 29567	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 29568	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 29569	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)))))
19.6.15  Positive operators (cont.)
Theorem	leopg 29570*	Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐵) → (𝑇 ≤op 𝑈 ↔ ((𝑈 −op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥))))
Theorem	leop 29571*	Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈 −op 𝑇)‘𝑥) ·ih 𝑥)))
Theorem	leop2 29572*	Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)))
Theorem	leop3 29573	Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ 0hop ≤op (𝑈 −op 𝑇)))
Theorem	leoppos 29574*	Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥)))
Theorem	leoprf2 29575	The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇: ℋ⟶ ℋ → 𝑇 ≤op 𝑇)
Theorem	leoprf 29576	The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 𝑇 ≤op 𝑇)
Theorem	leopsq 29577	The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ (𝑇 ∈ HrmOp → 0hop ≤op (𝑇 ∘ 𝑇))
Theorem	0leop 29578	The zero operator is a positive operator. (The literature calls it "positive", even though in some sense it is really "nonnegative".) Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ 0hop ≤op 0hop
Theorem	idleop 29579	The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
⊢ 0hop ≤op Iop
Theorem	leopadd 29580	The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈))
Theorem	leopmuli 29581	The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hop ≤op 𝑇)) → 0hop ≤op (𝐴 ·op 𝑇))
Theorem	leopmul 29582	The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 0 < 𝐴) → ( 0hop ≤op 𝑇 ↔ 0hop ≤op (𝐴 ·op 𝑇)))
Theorem	leopmul2i 29583	Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 𝑇 ≤op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))
Theorem	leoptri 29584	The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑇 ≤op 𝑈 ∧ 𝑈 ≤op 𝑇) ↔ 𝑇 = 𝑈))
Theorem	leoptr 29585	The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈)
Theorem	leopnmid 29586	A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ) → 𝑇 ≤op ((normop‘𝑇) ·op Iop ))
Theorem	nmopleid 29587	A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ HrmOp ∧ (normop‘𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop‘𝑇)) ·op 𝑇) ≤op Iop )
Theorem	opsqrlem1 29588*	Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (normop‘𝑇) ∈ ℝ    &   ⊢ 0hop ≤op 𝑇    &   ⊢ 𝑅 = ((1 / (normop‘𝑇)) ·op 𝑇)    &   ⊢ (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hop ≤op 𝑢 ∧ (𝑢 ∘ 𝑢) = 𝑅))    ⇒   ⊢ (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hop ≤op 𝑣 ∧ (𝑣 ∘ 𝑣) = 𝑇))
Theorem	opsqrlem2 29589*	Lemma for opsqri . 𝐹‘𝑁 is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ (𝐹‘1) = 0hop
Theorem	opsqrlem3 29590*	Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇 −op (𝐺 ∘ 𝐺)))))
Theorem	opsqrlem4 29591*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ 𝐹:ℕ⟶HrmOp
Theorem	opsqrlem5 29592*	Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁))))))
Theorem	opsqrlem6 29593*	Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥)))))    &   ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop }))    &   ⊢ 𝑇 ≤op Iop    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ≤op Iop )
19.6.16  Projectors as operators
Theorem	pjhmopi 29594	A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ HrmOp
Theorem	pjlnopi 29595	A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ LinOp
Theorem	pjnmopi 29596	The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (𝐻 ≠ 0ℋ → (normop‘(projℎ‘𝐻)) = 1)
Theorem	pjbdlni 29597	A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
⊢ 𝐻 ∈ Cℋ    ⇒   ⊢ (projℎ‘𝐻) ∈ BndLinOp
Theorem	pjhmop 29598	A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) ∈ HrmOp)
Theorem	hmopidmchi 29599	An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (𝑇 ∘ 𝑇) = 𝑇    ⇒   ⊢ ran 𝑇 ∈ Cℋ
Theorem	hmopidmpji 29600	An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 29599 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
⊢ 𝑇 ∈ HrmOp    &   ⊢ (𝑇 ∘ 𝑇) = 𝑇    ⇒   ⊢ 𝑇 = (projℎ‘ran 𝑇)