P. 287 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ipval2lem4 28601	Lemma for ipval3 28604. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℂ)
Theorem	ipval2 28602	Expansion of the inner product value ipval 28598. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4))
Theorem	4ipval2 28603	Four times the inner product value ipval3 28604, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴𝑃𝐵)) = ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))))
Theorem	ipval3 28604	Expansion of the inner product value ipval 28598. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝑀𝐵))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝑀(i𝑆𝐵)))↑2)))) / 4))
Theorem	ipidsq 28605	The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2))
Theorem	ipnm 28606	Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (√‘(𝐴𝑃𝐴)))
Theorem	dipcl 28607	An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
Theorem	ipf 28608	Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑃:(𝑋 × 𝑋)⟶ℂ)
Theorem	dipcj 28609	The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))
Theorem	ipipcj 28610	An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) · (𝐵𝑃𝐴)) = ((abs‘(𝐴𝑃𝐵))↑2))
Theorem	diporthcom 28611	Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0))
Theorem	dip0r 28612	Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝑍) = 0)
Theorem	dip0l 28613	Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝑃𝐴) = 0)
Theorem	ipz 28614	The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍))
Theorem	dipcn 28615	Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝐶 = (IndMet‘𝑈)    &   ⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
18.3.5  Subspaces
Syntax	css 28616	Extend class notation with the class of all subspaces of normed complex vector spaces.
class SubSp
Definition	df-ssp 28617*	Define the class of all subspaces of normed complex vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))})
Theorem	sspval 28618*	The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
Theorem	isssp 28619	The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝐹 = ( +𝑣 ‘𝑊)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑀 = (normCV‘𝑊)    &   ⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))
Theorem	sspid 28620	A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
⊢ 𝐻 = (SubSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻)
Theorem	sspnv 28621	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 28622	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 28623	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 28624	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 28625	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 28626	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 28627*	Lemma for sspm 28629 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 28628	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 28629	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 28630	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 28631	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 28632	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 28633	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 28634	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 28635	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
Syntax	cnmoo 28636	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD
Syntax	cblo 28637	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
Syntax	c0o 28638	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
Definition	df-lno 28639*	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 28640*	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 28641*	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 28642*	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 28643	Adjoint of an operator.
class adj
Syntax	chmo 28644	Set of Hermitional (self-adjoint) operators.
class HmOp
Definition	df-aj 28645*	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 28646*	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 28647*	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 28648*	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 28649	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 28650	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 28651	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 28652	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 28653	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 28654	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 28655	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 28656	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 28657*	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 28658*	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 28659*	The set in the supremum of the operator norm definition df-nmoo 28640 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 28660*	The set in the supremum of the operator norm definition df-nmoo 28640 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 28661	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 28662	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 28663	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 28664	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 28665	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 28666	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 28667*	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 28668*	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 28669*	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 28670*	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 28671*	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 28672*	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 28673*	Alternate shorter proof of nmounbseqi 28672 based on Axioms ax-reg 9102 and ax-ac2 9936 instead of ax-cc 9908. (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 28674*	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 28675*	Alternate shorter proof of nmobndseqi 28674 based on Axioms ax-reg 9102 and ax-ac2 9936 instead of ax-cc 9908. (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 28676*	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 28677	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 28678	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 28679	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 28680	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 28681	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 28682	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 28683	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 28684	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 28685	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 28686	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 28687	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 28688	Lemma for nmlno0i 28689. (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 28689	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 28690	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 28691*	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 28692	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 28693*	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 28694	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 28695	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 28696*	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 28697*	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 28698	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 28699	Lemma for blocni 28700 and lnocni 28701. 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 28700	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)