P. 327 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lnophmlem2 32601*	Lemma for lnophmi 32602. (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 32602*	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 32603*	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 32604	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 32605	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 32606	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 32607	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 32608	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 32609	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 32610*	Lemma for nmcopexi 32611 and nmcfnexi 32635. 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 32611	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 32612	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 32613	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 32614	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 32615	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 32616	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 32617*	Lemma for lnopconi 32618 and lnfnconi 32639. (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 32618*	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 32619*	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 32620	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 32621	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 32622	A continuous linear operator is a bounded linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
⊢ (LinOp ∩ ContOp) = BndLinOp
Theorem	lnopcnre 32623	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 32624	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 32625	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 32626	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 32627	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 32628	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 32629	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 32630	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 32631	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 32632	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 32633	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 32634	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 32635	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 32636	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 32637	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 32638	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 32639*	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 32640*	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 32641	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 32642	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 32643	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 32644	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 32645	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
20.6.11  Riesz lemma
Theorem	riesz3i 32646*	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 32647*	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 32648*	A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104. See riesz2 32650 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 32649*	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 32650. For the continuous linear functional version, see riesz3i 32646 and riesz4 32648. (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 32650*	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 32649. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → ∃!𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑥 ·ih 𝑦))
20.6.12  Adjoints (cont.)
Theorem	cnlnadjlem1 32651*	Lemma for cnlnadji 32660 (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 32652*	Lemma for cnlnadji 32660. 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 32653*	Lemma for cnlnadji 32660. By riesz4 32648, 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 32654*	Lemma for cnlnadji 32660. 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 32655*	Lemma for cnlnadji 32660. 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 32656*	Lemma for cnlnadji 32660. 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 32657*	Lemma for cnlnadji 32660. 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 32658*	Lemma for cnlnadji 32660. 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 32659*	Lemma for cnlnadji 32660. 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 32660*	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 32661*	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 32662*	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 32663*	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 32664	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 32665	Every bounded linear Hilbert space operator has an adjoint. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
⊢ BndLinOp ⊆ dom adjℎ
Theorem	bdopadj 32666	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 32667	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 32668	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 32669	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 32670	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 32671	Every operator with an adjoint is linear. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)
⊢ dom adjℎ ⊆ LinOp
Theorem	nmopadjlei 32672	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 32673	Lemma for nmopadji 32674. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
⊢ 𝑇 ∈ BndLinOp    ⇒   ⊢ (normop‘(adjℎ‘𝑇)) ≤ (normop‘𝑇)
Theorem	nmopadji 32674	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 32675	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 32676	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 32677	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 32678	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 32679	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 32680	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 32681	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 32682	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 32683	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 32684	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 32685	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 32686	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 32687	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 )
20.6.13  Quantum computation error bound theorem
Theorem	unierri 32688	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)))
20.6.14  Dirac bra-ket notation (cont.)
Theorem	branmfn 32689	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 32690	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 32691	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 32692	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 32693	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 32694*	Value of the converse of the bra function. Based on the Riesz Lemma riesz4 32648, this very important theorem not only justifies the Dirac bra-ket notation, but allows 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 32695	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 32696	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 32697	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 32698*	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 32699	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 32700	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))