Home Metamath Proof ExplorerTheorem List (p. 276 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnvop 27501 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

18.3.2  Examples of normed complex vector spaces

Theoremcnnv 27502 The set of complex numbers is a normed complex vector space. The vector operation is +, the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       𝑈 ∈ NrmCVec

Theoremcnnvg 27503 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        + = ( +𝑣𝑈)

Theoremcnnvba 27504 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       ℂ = (BaseSet‘𝑈)

Theoremcnnvs 27505 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        · = ( ·𝑠OLD𝑈)

Theoremcnnvnm 27506 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       abs = (normCV𝑈)

Theoremcnnvm 27507 The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        − = ( −𝑣𝑈)

Theoremelimnv 27508 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑈 ∈ NrmCVec       if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Theoremelimnvu 27509 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ NrmCVec

18.3.3  Induced metric of a normed complex vector space

Theoremimsval 27510 Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))

Theoremimsdval 27511 Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Theoremimsdval2 27512 Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))

Theoremnvnd 27513 The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐷𝑍))

Theoremimsdf 27514 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ)

Theoremimsmetlem 27515 Lemma for imsmet 27516. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = (inv‘𝐺)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝑈 ∈ NrmCVec       𝐷 ∈ (Met‘𝑋)

Theoremimsmet 27516 The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋))

Theoremimsxmet 27517 The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 ∈ (∞Met‘𝑋))

Theoremcnims 27518 The metric induced on the complex numbers. cnmet 22556 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩    &   𝐷 = (abs ∘ − )       𝐷 = (IndMet‘𝑈)

Theoremvacn 27519 Vector addition is jointly continuous in both arguments. (Contributed by Jeff Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Theoremnmcvcn 27520 The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
𝑁 = (normCV𝑈)    &   𝐶 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (topGen‘ran (,))       (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾))

Theoremnmcnc 27521 The norm of a normed complex vector space is a continuous function to . (For , see nmcvcn 27520.) (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 27522* Lemma for smcn 27523. (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 27523 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 27524 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 27525 Extend class notation with the class inner product functions.
class ·𝑖OLD

Definitiondf-dip 27526* 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 27527* 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 27528* 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 27529 Lemma for ipval3 27534. (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 27530 Lemma for ipval3 27534. (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 27531 Lemma for ipval3 27534. (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 27532 Expansion of the inner product value ipval 27528. (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 27533 Four times the inner product value ipval3 27534, 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 27534 Expansion of the inner product value ipval 27528. (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 27535 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 27536 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 27537 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 27538 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 27539 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 27540 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 27541 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 27542 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 27543 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 27544 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 27545 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 27546 Extend class notation with the class of all subspaces of normed complex vector spaces.
class SubSp

Definitiondf-ssp 27547* 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 27548* 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 27549 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 27550 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 27551 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 27552 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 27553 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 27554 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 27555 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 27556 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 27557* Lemma for sspm 27559 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 27558 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 27559 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 27560 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 27561 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 27562 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 27563 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 27564 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 27565 Extend class notation with the class of linear operators on normed complex vector spaces.
class LnOp

Syntaxcnmoo 27566 Extend class notation with the class of operator norms on normed complex vector spaces.
class normOpOLD

Syntaxcblo 27567 Extend class notation with the class of bounded linear operators on normed complex vector spaces.
class BLnOp

Syntaxc0o 27568 Extend class notation with the class of zero operators on normed complex vector spaces.
class 0op

Definitiondf-lno 27569* 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 27570* 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 27571* 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 27572* 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 27573 Adjoint of an operator.

Syntaxchmo 27574 Set of Hermitional (self-adjoint) operators.
class HmOp

Definitiondf-aj 27575* 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 27576* 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 27577* 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 27578* 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 27579 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 27580 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 27581 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 27582 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 27583 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 27584 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 27585 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 27586 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 27587* 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 27588* 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 27589* The set in the supremum of the operator norm definition df-nmoo 27570 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 27590* The set in the supremum of the operator norm definition df-nmoo 27570 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 27591 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 27592 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 27593 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 27594 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 27595 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 27596 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 27597* 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 27598* 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 27599* 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 27600* 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)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >