P. 313 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dipcl 31201	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 31202	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 31203	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 31204	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 31205	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 31206	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 31207	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 31208	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 31209	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))
19.3.5  Subspaces
Syntax	css 31210	Extend class notation with the class of all subspaces of normed complex vector spaces.
class SubSp
Definition	df-ssp 31211*	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 31212*	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 31213	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 31214	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 31215	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 31216	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 31217	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 31218	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 31219	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 31220	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 31221*	Lemma for sspm 31223 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 31222	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 31223	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 31224	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 31225	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 31226	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 31227	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 31228	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 31229	Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
Syntax	cnmoo 31230	Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD
Syntax	cblo 31231	Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
Syntax	c0o 31232	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
Definition	df-lno 31233*	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 31234*	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 31235*	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 31236*	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 31237	Adjoint of an operator.
class adj
Syntax	chmo 31238	Set of Hermitional (self-adjoint) operators.
class HmOp
Definition	df-aj 31239*	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 31240*	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 31241*	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 31242*	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 31243	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 31244	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 31245	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 31246	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 31247	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 31248	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 31249	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 31250	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 31251*	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 31252*	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 31253*	The set in the supremum of the operator norm definition df-nmoo 31234 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 31254*	The set in the supremum of the operator norm definition df-nmoo 31234 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 31255	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 31256	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 31257	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 31258	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 31259	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 31260	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 31261*	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 31262*	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 31263*	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 31264*	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 31265*	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 31266*	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 31267*	Alternate shorter proof of nmounbseqi 31266 based on Axioms ax-reg 9568 and ax-ac2 10469 instead of ax-cc 10441. (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 31268*	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 31269*	Alternate shorter proof of nmobndseqi 31268 based on Axioms ax-reg 9568 and ax-ac2 10469 instead of ax-cc 10441. (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 31270*	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 31271	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 31272	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 31273	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 31274	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 31275	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 31276	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 31277	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 31278	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 31279	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 31280	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 31281	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 31282	Lemma for nmlno0i 31283. (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 31283	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 31284	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 31285*	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 31286	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 31287*	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 31288	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 31289	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 31290*	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 31291*	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 31292	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 31293	Lemma for blocni 31294 and lnocni 31295. 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 31294	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 31295	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 31296	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 31297	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 31298*	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 31299*	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 31300	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)))