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

Theoremipval2lem4 28601 Lemma for ipval3 28604. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       (((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℂ)

Theoremipval2 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4))

Theorem4ipval2 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (4 · (𝐴𝑃𝐵)) = ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))))

Theoremipval3 28604 Expansion of the inner product value ipval 28598. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝑀𝐵))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝑀(i𝑆𝐵)))↑2)))) / 4))

Theoremipidsq 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 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝑃𝐴) = ((𝑁𝐴)↑2))

Theoremipnm 28606 Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) = (√‘(𝐴𝑃𝐴)))

Theoremdipcl 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 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) ∈ ℂ)

Theoremipf 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 = (·𝑖OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑃:(𝑋 × 𝑋)⟶ℂ)

Theoremdipcj 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 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴))

Theoremipipcj 28610 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑃𝐵) · (𝐵𝑃𝐴)) = ((abs‘(𝐴𝑃𝐵))↑2))

Theoremdiporthcom 28611 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0))

Theoremdip0r 28612 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝑃𝑍) = 0)

Theoremdip0l 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 = (0vec𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝑃𝐴) = 0)

Theoremipz 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 = (0vec𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍))

Theoremdipcn 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 = (·𝑖OLD𝑈)    &   𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (TopOpen‘ℂfld)       (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))

18.3.5  Subspaces

Syntaxcss 28616 Extend class notation with the class of all subspaces of normed complex vector spaces.
class SubSp

Definitiondf-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 ∣ (( +𝑣𝑤) ⊆ ( +𝑣𝑢) ∧ ( ·𝑠OLD𝑤) ⊆ ( ·𝑠OLD𝑢) ∧ (normCV𝑤) ⊆ (normCV𝑢))})

Theoremsspval 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝐻 = (SubSp‘𝑈)       (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})

Theoremisssp 28619 The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝐹 = ( +𝑣𝑊)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑅 = ( ·𝑠OLD𝑊)    &   𝑁 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝐻 = (SubSp‘𝑈)       (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))

Theoremsspid 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 → 𝑈𝐻)

Theoremsspnv 28621 A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑊 ∈ NrmCVec)

Theoremsspba 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 ∧ 𝑊𝐻) → 𝑌𝑋)

Theoremsspg 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 = ( +𝑣𝑈)    &   𝐹 = ( +𝑣𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Theoremsspgval 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 = ( +𝑣𝑈)    &   𝐹 = ( +𝑣𝑊)    &   𝐻 = (SubSp‘𝑈)       (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴𝑌𝐵𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Theoremssps 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 = ( ·𝑠OLD𝑈)    &   𝑅 = ( ·𝑠OLD𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))

Theoremsspsval 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 = ( ·𝑠OLD𝑈)    &   𝑅 = ( ·𝑠OLD𝑊)    &   𝐻 = (SubSp‘𝑈)       (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))

Theoremsspmlem 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 ∧ 𝑊𝐻) ∧ (𝑥𝑌𝑦𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    &   (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅)    &   (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Theoremsspmval 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 = ( −𝑣𝑈)    &   𝐿 = ( −𝑣𝑊)    &   𝐻 = (SubSp‘𝑈)       (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴𝑌𝐵𝑌)) → (𝐴𝐿𝐵) = (𝐴𝑀𝐵))

Theoremsspm 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 = ( −𝑣𝑈)    &   𝐿 = ( −𝑣𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐿 = (𝑀 ↾ (𝑌 × 𝑌)))

Theoremsspz 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 = (0vec𝑈)    &   𝑄 = (0vec𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑄 = 𝑍)

Theoremsspn 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑀 = (𝑁𝑌))

Theoremsspnval 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻𝐴𝑌) → (𝑀𝐴) = (𝑁𝐴))

Theoremsspimsval 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 ∧ 𝑊𝐻) ∧ (𝐴𝑌𝐵𝑌)) → (𝐴𝐶𝐵) = (𝐴𝐷𝐵))

Theoremsspims 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 ∧ 𝑊𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌)))

18.4  Operators on complex vector spaces

18.4.1  Definitions and basic properties

Syntaxclno 28635 Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp

Syntaxcnmoo 28636 Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD

Syntaxcblo 28637 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp

Syntaxc0o 28638 Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op

Definitiondf-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( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))})

Definitiondf-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)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))

Definitiondf-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) < +∞})

Definitiondf-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) × {(0vec𝑤)}))

Syntaxcaj 28643 Adjoint of an operator.

Syntaxchmo 28644 Set of Hermitional (self-adjoint) operators.
class HmOp

Definitiondf-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)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})

Definitiondf-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})

Theoremlnoval 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 = ( +𝑣𝑈)    &   𝐻 = ( +𝑣𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})

Theoremislno 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 = ( +𝑣𝑈)    &   𝐻 = ( +𝑣𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))))

Theoremlnolin 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 = ( +𝑣𝑈)    &   𝐻 = ( +𝑣𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))

Theoremlnof 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 ∧ 𝑇𝐿) → 𝑇:𝑋𝑌)

Theoremlno0 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 = (0vec𝑈)    &   𝑍 = (0vec𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → (𝑇𝑄) = 𝑍)

Theoremlnocoi 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    &   𝑆𝐿    &   𝑇𝑀       (𝑇𝑆) ∈ 𝑁

Theoremlnoadd 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 = ( +𝑣𝑈)    &   𝐻 = ( +𝑣𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇𝐴)𝐻(𝑇𝐵)))

Theoremlnosub 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 = ( −𝑣𝑈)    &   𝑁 = ( −𝑣𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴𝑋𝐵𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = ((𝑇𝐴)𝑁(𝑇𝐵)))

Theoremlnomul 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 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋)) → (𝑇‘(𝐴𝑅𝐵)) = (𝐴𝑆(𝑇𝐵)))

Theoremnvo00 28656 Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍}))

Theoremnmoofval 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))

Theoremnmooval 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))

Theoremnmosetre 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 = (normCV𝑊)       ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))} ⊆ ℝ)

Theoremnmosetn0 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 = (0vec𝑈)    &   𝑀 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (𝑁‘(𝑇𝑍)) ∈ {𝑥 ∣ ∃𝑦𝑋 ((𝑀𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑦)))})

Theoremnmoxr 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:𝑋𝑌) → (𝑁𝑇) ∈ ℝ*)

Theoremnmooge0 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:𝑋𝑌) → 0 ≤ (𝑁𝑇))

Theoremnmorepnf 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:𝑋𝑌) → ((𝑁𝑇) ∈ ℝ ↔ (𝑁𝑇) ≠ +∞))

Theoremnmoreltpnf 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:𝑋𝑌) → ((𝑁𝑇) ∈ ℝ ↔ (𝑁𝑇) < +∞))

Theoremnmogtmnf 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:𝑋𝑌) → -∞ < (𝑁𝑇))

Theoremnmoolb 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑀‘(𝑇𝐴)) ≤ (𝑁𝑇))

Theoremnmoubi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))

Theoremnmoub3i 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌𝐴 ∈ ℝ ∧ ∀𝑥𝑋 (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐿𝑥))) → (𝑁𝑇) ≤ (abs‘𝐴))

Theoremnmoub2i 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥𝑋 (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐿𝑥))) → (𝑁𝑇) ≤ 𝐴)

Theoremnmobndi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑟)))

Theoremnmounbi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇:𝑋𝑌 → ((𝑁𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇𝑦)))))

Theoremnmounbseqi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝑁𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓𝑘))))))

TheoremnmounbseqiALT 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝑁𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓𝑘))))))

Theoremnmobndseqi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)

TheoremnmobndseqiALT 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)

Theorembloval 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 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})

Theoremisblo 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) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))

Theoremisblo2 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) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) ∈ ℝ)))

Theorembloln 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 ∧ 𝑇𝐵) → 𝑇𝐿)

Theoremblof 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 ∧ 𝑇𝐵) → 𝑇:𝑋𝑌)

Theoremnmblore 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 ∧ 𝑇𝐵) → (𝑁𝑇) ∈ ℝ)

Theorem0ofval 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 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Theorem0oval 28683 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = 𝑍)

Theorem0oo 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:𝑋𝑌)

Theorem0lno 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) → 𝑍𝐿)

Theoremnmoo0 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) → (𝑁𝑍) = 0)

Theorem0blo 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) → 𝑍𝐵)

Theoremnmlno0lem 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    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝑃 = (0vec𝑈)    &   𝑄 = (0vec𝑊)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)       ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍)

Theoremnmlno0i 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       (𝑇𝐿 → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlno0 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 ∧ 𝑇𝐿) → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlnoubi 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 = (0vec𝑈)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥𝑋 (𝑥𝑍 → (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐾𝑥)))) → (𝑁𝑇) ≤ 𝐴)

Theoremnmlnogt0 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 ∧ 𝑇𝐿) → (𝑇𝑍 ↔ 0 < (𝑁𝑇)))

Theoremlnon0 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 = (0vec𝑈)    &   𝑂 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ 𝑇𝑂) → ∃𝑥𝑋 𝑥𝑍)

Theoremnmblolbii 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐵       (𝐴𝑋 → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremnmblolbi 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐵𝐴𝑋) → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremisblo3i 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 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇𝐵 ↔ (𝑇𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (𝑀𝑦))))

Theoremblo3i 28697* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿𝐴 ∈ ℝ ∧ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝐴 · (𝑀𝑦))) → 𝑇𝐵)

Theoremblometi 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       ((𝑇𝐵𝑃𝑋𝑄𝑋) → ((𝑇𝑃)𝐷(𝑇𝑄)) ≤ ((𝑁𝑇) · (𝑃𝐶𝑄)))

Theoremblocnilem 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    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)       ((𝑃𝑋𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇𝐵)

Theoremblocni 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    &   𝑇𝐿       (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇𝐵)

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