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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmcnc 28401 The norm of a normed complex vector space is a continuous function to . (For , see nmcvcn 28400.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
𝑁 = (normCV𝑈)    &   𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (TopOpen‘ℂfld)       (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾))
 
Theoremsmcnlem 28402* Lemma for smcn 28403. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝐾 = (TopOpen‘ℂfld)    &   𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑈 ∈ NrmCVec    &   𝑇 = (1 / (1 + ((((𝑁𝑦) + (abs‘𝑥)) + 1) / 𝑟)))       𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)
 
Theoremsmcn 28403 Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝐾 = (TopOpen‘ℂfld)       (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
 
Theoremvmcn 28404 Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝑀 = ( −𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
 
18.3.4  Inner product
 
Syntaxcdip 28405 Extend class notation with the class inner product functions.
class ·𝑖OLD
 
Definitiondf-dip 28406* Define a function that maps a normed complex vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st𝑤), the scalar product is (2nd𝑤), and the norm is 𝑛. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
 
Theoremdipfval 28407* The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑃 = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
 
Theoremipval 28408* Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, the norm is 𝑁, and the set of vectors is 𝑋. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
 
Theoremipval2lem2 28409 Lemma for ipval3 28414. (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) ∈ ℝ)
 
Theoremipval2lem3 28410 Lemma for ipval3 28414. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ)
 
Theoremipval2lem4 28411 Lemma for ipval3 28414. (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 28412 Expansion of the inner product value ipval 28408. (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 28413 Four times the inner product value ipval3 28414, 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 28414 Expansion of the inner product value ipval 28408. (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 28415 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 28416 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 28417 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 28418 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 28419 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 28420 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 28421 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 28422 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 28423 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 28424 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 28425 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 28426 Extend class notation with the class of all subspaces of normed complex vector spaces.
class SubSp
 
Definitiondf-ssp 28427* 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 28428* 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 28429 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 28430 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 28431 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 28432 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 28433 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 28434 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 28435 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 28436 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 28437* Lemma for sspm 28439 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 28438 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 28439 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 28440 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 28441 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 28442 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 28443 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 28444 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 28445 Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp
 
Syntaxcnmoo 28446 Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD
 
Syntaxcblo 28447 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp
 
Syntaxc0o 28448 Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op
 
Definitiondf-lno 28449* 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 28450* 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 28451* 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 28452* 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 28453 Adjoint of an operator.
class adj
 
Syntaxchmo 28454 Set of Hermitional (self-adjoint) operators.
class HmOp
 
Definitiondf-aj 28455* 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 28456* 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 28457* 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 28458* 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 28459 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 28460 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 28461 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 28462 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 28463 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 28464 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 28465 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 28466 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 28467* 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 28468* 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 28469* The set in the supremum of the operator norm definition df-nmoo 28450 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 28470* The set in the supremum of the operator norm definition df-nmoo 28450 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 28471 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 28472 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 28473 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 28474 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 28475 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 28476 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 28477* 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 28478* 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 28479* 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 28480* 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 28481* 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 28482* 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 28483* Alternate shorter proof of nmounbseqi 28482 based on axioms ax-reg 9045 and ax-ac2 9874 instead of ax-cc 9846. (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 28484* 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 28485* Alternate shorter proof of nmobndseqi 28484 based on axioms ax-reg 9045 and ax-ac2 9874 instead of ax-cc 9846. (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 28486* 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 28487 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 28488 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 28489 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 28490 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 28491 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 28492 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 28493 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 28494 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 28495 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 28496 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 28497 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 28498 Lemma for nmlno0i 28499. (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 28499 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 28500 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))
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