P. 292 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 29101-29200 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ssps 29101	Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

Theorem	sspsval 29102	Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))

Theorem	sspmlem 29103*	Lemma for sspm 29105 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) & ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) & ⊢ (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅) & ⊢ (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Theorem	sspmval 29104	Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝐿 = ( −_𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐿𝐵) = (𝐴𝑀𝐵))

Theorem	sspm 29105	Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝐿 = ( −_𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐿 = (𝑀 ↾ (𝑌 × 𝑌)))

Theorem	sspz 29106	The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝑄 = (0_vec‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍)

Theorem	sspn 29107	The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))

Theorem	sspnval 29108	The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))

Theorem	sspimsval 29109	The induced metric on a subspace in terms of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐶𝐵) = (𝐴𝐷𝐵))

Theorem	sspims 29110	The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌)))

18.4 Operators on complex vector spaces

18.4.1 Definitions and basic properties

Syntax	clno 29111	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp

Syntax	cnmoo 29112	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOp_OLD

Syntax	cblo 29113	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp

Syntax	c0o 29114	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0_op

Definition	df-lno 29115*	Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e., the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑_m (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·_𝑠OLD ‘𝑢)𝑦)( +_𝑣 ‘𝑢)𝑧)) = ((𝑥( ·_𝑠OLD ‘𝑤)(𝑡‘𝑦))( +_𝑣 ‘𝑤)(𝑡‘𝑧))})

Definition	df-nmoo 29116*	Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces ⟨𝑢, 𝑤⟩. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ normOp_OLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑_m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((norm_CV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑤)‘(𝑡‘𝑧)))}, ℝ^*, < )))

Definition	df-blo 29117*	Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOp_OLD 𝑤)‘𝑡) < +∞})

Definition	df-0o 29118*	Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 0_op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0_vec‘𝑤)}))

Syntax	caj 29119	Adjoint of an operator.
class adj

Syntax	chmo 29120	Set of Hermitional (self-adjoint) operators.
class HmOp

Definition	df-aj 29121*	Define the adjoint of an operator (if it exists). The domain of 𝑈adj𝑊 is the set of all operators from 𝑈 to 𝑊 that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that 𝑈 and 𝑊 be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·_𝑖OLD‘𝑤)𝑦) = (𝑥(·_𝑖OLD‘𝑢)(𝑠‘𝑦)))})

Definition	df-hmo 29122*	Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡})

Theorem	lnoval 29123*	The set of linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝐻 = ( +_𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌 ↑_m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))})

Theorem	islno 29124*	The predicate "is a linear operator." (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝐻 = ( +_𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧)))))

Theorem	lnolin 29125	Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝐻 = ( +_𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))

Theorem	lnof 29126	A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)

Theorem	lno0 29127	The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑄 = (0_vec‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍)

Theorem	lnocoi 29128	The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑀 = (𝑊 LnOp 𝑋) & ⊢ 𝑁 = (𝑈 LnOp 𝑋) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑋 ∈ NrmCVec & ⊢ 𝑆 ∈ 𝐿 & ⊢ 𝑇 ∈ 𝑀 ⇒ ⊢ (𝑇 ∘ 𝑆) ∈ 𝑁

Theorem	lnoadd 29129	Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝐻 = ( +_𝑣 ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵)))

Theorem	lnosub 29130	Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = ( −_𝑣 ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = ((𝑇‘𝐴)𝑁(𝑇‘𝐵)))

Theorem	lnomul 29131	Scalar multiplication property of a linear operator. (Contributed by NM, 5-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑅𝐵)) = (𝐴𝑆(𝑇‘𝐵)))

Theorem	nvo00 29132	Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍}))

Theorem	nmoofval 29133*	The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑_m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ^*, < )))

Theorem	nmooval 29134*	The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))

Theorem	nmosetre 29135*	The set in the supremum of the operator norm definition df-nmoo 29116 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (norm_CV‘𝑊) ⇒ ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ)

Theorem	nmosetn0 29136*	The set in the supremum of the operator norm definition df-nmoo 29116 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))})

Theorem	nmoxr 29137	The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)

Theorem	nmooge0 29138	The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇))

Theorem	nmorepnf 29139	The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))

Theorem	nmoreltpnf 29140	The norm of any operator is real iff it is less than plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))

Theorem	nmogtmnf 29141	The norm of an operator is greater than minus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))

Theorem	nmoolb 29142	A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇))

Theorem	nmoubi 29143*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

Theorem	nmoub3i 29144*	An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))

Theorem	nmoub2i 29145*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ 𝐴)

Theorem	nmobndi 29146*	Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))

Theorem	nmounbi 29147*	Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))

Theorem	nmounbseqi 29148*	An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘))))))

Theorem	nmounbseqiALT 29149*	Alternate shorter proof of nmounbseqi 29148 based on Axioms ax-reg 9360 and ax-ac2 10228 instead of ax-cc 10200. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘))))))

Theorem	nmobndseqi 29150*	A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ)

Theorem	nmobndseqiALT 29151*	Alternate shorter proof of nmobndseqi 29150 based on Axioms ax-reg 9360 and ax-ac2 10228 instead of ax-cc 10200. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ)

Theorem	bloval 29152*	The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

Theorem	isblo 29153	The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))

Theorem	isblo2 29154	The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))

Theorem	bloln 29155	A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿)

Theorem	blof 29156	A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌)

Theorem	nmblore 29157	The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

Theorem	0ofval 29158	The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑊) & ⊢ 𝑂 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Theorem	0oval 29159	Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑊) & ⊢ 𝑂 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

Theorem	0oo 29160	The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌)

Theorem	0lno 29161	The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Theorem	nmoo0 29162	The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0)

Theorem	0blo 29163	The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐵)

Theorem	nmlno0lem 29164	Lemma for nmlno0i 29165. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝑃 = (0_vec‘𝑈) & ⊢ 𝑄 = (0_vec‘𝑊) & ⊢ 𝐾 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) ⇒ ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)

Theorem	nmlno0i 29165	The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))

Theorem	nmlno0 29166	The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))

Theorem	nmlnoubi 29167*	An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝐾 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ≠ 𝑍 → (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐾‘𝑥)))) → (𝑁‘𝑇) ≤ 𝐴)

Theorem	nmlnogt0 29168	The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇)))

Theorem	lnon0 29169*	The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝑂 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍)

Theorem	nmblolbii 29170	A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Theorem	nmblolbi 29171	A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Theorem	isblo3i 29172*	The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦))))

Theorem	blo3i 29173*	Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵)

Theorem	blometi 29174	Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) ≤ ((𝑁‘𝑇) · (𝑃𝐶𝑄)))

Theorem	blocnilem 29175	Lemma for blocni 29176 and lnocni 29177. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵)

Theorem	blocni 29176	A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 ⇒ ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)

Theorem	lnocni 29177	If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.)
⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾))

Theorem	blocn 29178	A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵))

Theorem	blocn2 29179	A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.)
⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾))

Theorem	ajfval 29180*	The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑄 = (·_𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})

Theorem	hmoval 29181*	The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡})

Theorem	ishmo 29182	The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇)))

18.5 Inner product (pre-Hilbert) spaces

18.5.1 Definition and basic properties

Syntax	ccphlo 29183	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil_OLD

Definition	df-ph 29184*	Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is 𝑔, the scalar product is 𝑠, and the norm is 𝑛. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ CPreHil_OLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})

Theorem	phnv 29185	Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)

Theorem	phrel 29186	The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ Rel CPreHil_OLD

Theorem	phnvi 29187	Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ 𝑈 ∈ NrmCVec

Theorem	isphg 29188*	The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHil_OLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Theorem	phop 29189	A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

18.5.2 Examples of pre-Hilbert spaces

Theorem	cncph 29190	The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩ ⇒ ⊢ 𝑈 ∈ CPreHil_OLD

Theorem	elimph 29191	Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋

Theorem	elimphu 29192	Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.)
⊢ if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHil_OLD

18.5.3 Properties of pre-Hilbert spaces

Theorem	isph 29193*	The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))

Theorem	phpar2 29194	The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Theorem	phpar 29195	The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Theorem	ip0i 29196	A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where 𝐽 is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2)))

Theorem	ip1ilem 29197	Lemma for ip1i 29198. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theorem	ip1i 29198	Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theorem	ip2i 29199	Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵))

Theorem	ipdirilem 29200	Lemma for ipdiri 29201. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

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20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46532

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