P. 308 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sspnv 30701	A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
Theorem	sspba 30702	The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋)
Theorem	sspg 30703	Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝐹 = ( +𝑣 ‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
Theorem	sspgval 30704	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)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝐹 = ( +𝑣 ‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Theorem	ssps 30705	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 30706	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 30707*	Lemma for sspm 30709 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 30708	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 30709	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 30710	The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑄 = (0vec‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍)
Theorem	sspn 30711	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌))
Theorem	sspnval 30712	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴))
Theorem	sspimsval 30713	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 30714	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)))
19.4  Operators on complex vector spaces
19.4.1  Definitions and basic properties
Syntax	clno 30715	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
Syntax	cnmoo 30716	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD
Syntax	cblo 30717	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
Syntax	c0o 30718	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
Definition	df-lno 30719*	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 30720*	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.)
⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
Definition	df-blo 30721*	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 normOpOLD 𝑤)‘𝑡) < +∞})
Definition	df-0o 30722*	Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)}))
Syntax	caj 30723	Adjoint of an operator.
class adj
Syntax	chmo 30724	Set of Hermitional (self-adjoint) operators.
class HmOp
Definition	df-aj 30725*	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 30726*	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 30727*	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 30728*	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 30729	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 30730	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 30731	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 = (0vec‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍)
Theorem	lnocoi 30732	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 30733	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 30734	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 30735	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 30736	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 30737*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
Theorem	nmooval 30738*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < ))
Theorem	nmosetre 30739*	The set in the supremum of the operator norm definition df-nmoo 30720 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝑁 = (normCV‘𝑊)    ⇒   ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ)
Theorem	nmosetn0 30740*	The set in the supremum of the operator norm definition df-nmoo 30720 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))})
Theorem	nmoxr 30741	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 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*)
Theorem	nmooge0 30742	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 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇))
Theorem	nmorepnf 30743	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 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))
Theorem	nmoreltpnf 30744	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 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))
Theorem	nmogtmnf 30745	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 normOpOLD 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))
Theorem	nmoolb 30746	A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    ⇒   ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇))
Theorem	nmoubi 30747*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))
Theorem	nmoub3i 30748*	An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))
Theorem	nmoub2i 30749*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30750*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
Theorem	nmounbi 30751*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
Theorem	nmounbseqi 30752*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30753*	Alternate shorter proof of nmounbseqi 30752 based on Axioms ax-reg 9478 and ax-ac2 10351 instead of ax-cc 10323. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30754*	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30755*	Alternate shorter proof of nmobndseqi 30754 based on Axioms ax-reg 9478 and ax-ac2 10351 instead of ax-cc 10323. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝐿 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30756*	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 normOpOLD 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})
Theorem	isblo 30757	The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))
Theorem	isblo2 30758	The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))
Theorem	bloln 30759	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 30760	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 30761	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 normOpOLD 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)
Theorem	0ofval 30762	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 = (0vec‘𝑊)    &   ⊢ 𝑂 = (𝑈 0op 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
Theorem	0oval 30763	Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑊)    &   ⊢ 𝑂 = (𝑈 0op 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)
Theorem	0oo 30764	The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌)
Theorem	0lno 30765	The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)
Theorem	nmoo0 30766	The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0)
Theorem	0blo 30767	The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐵)
Theorem	nmlno0lem 30768	Lemma for nmlno0i 30769. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    &   ⊢ 𝑇 ∈ 𝐿    &   ⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊)    &   ⊢ 𝑃 = (0vec‘𝑈)    &   ⊢ 𝑄 = (0vec‘𝑊)    &   ⊢ 𝐾 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    ⇒   ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)
Theorem	nmlno0i 30769	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 normOpOLD 𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
Theorem	nmlno0 30770	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 normOpOLD 𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))
Theorem	nmlnoubi 30771*	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 = (0vec‘𝑈)    &   ⊢ 𝐾 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 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 30772	The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝑍 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇)))
Theorem	lnon0 30773*	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 = (0vec‘𝑈)    &   ⊢ 𝑂 = (𝑈 0op 𝑊)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑊)    ⇒   ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍)
Theorem	nmblolbii 30774	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    &   ⊢ 𝑇 ∈ 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))
Theorem	nmblolbi 30775	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 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑊)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))
Theorem	isblo3i 30776*	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 = (normCV‘𝑈)    &   ⊢ 𝑁 = (normCV‘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 30777*	Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑈)    &   ⊢ 𝑁 = (normCV‘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 30778	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 normOpOLD 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 30779	Lemma for blocni 30780 and lnocni 30781. 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 30780	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 30781	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 30782	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 30783	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 30784*	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 30785*	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 30786	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)))
19.5  Inner product (pre-Hilbert) spaces
19.5.1  Definition and basic properties
Syntax	ccphlo 30787	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHilOLD
Definition	df-ph 30788*	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.)
⊢ CPreHilOLD = (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 30789	Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
Theorem	phrel 30790	The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ Rel CPreHilOLD
Theorem	phnvi 30791	Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 ∈ CPreHilOLD    ⇒   ⊢ 𝑈 ∈ NrmCVec
Theorem	isphg 30792*	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⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
Theorem	phop 30793	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 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
19.5.2  Examples of pre-Hilbert spaces
Theorem	cncph 30794	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 ∈ CPreHilOLD
Theorem	elimph 30795	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 = (0vec‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    ⇒   ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋
Theorem	elimphu 30796	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 ∈ CPreHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHilOLD
19.5.3  Properties of pre-Hilbert spaces
Theorem	isph 30797*	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 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
Theorem	phpar2 30798	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 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
Theorem	phpar 30799	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 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
Theorem	ip0i 30800	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 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    &   ⊢ 𝐶 ∈ 𝑋    &   ⊢ 𝑁 = (normCV‘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)))