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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nvcli 26901 | The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝐴 ∈ 𝑋 ⇒ ⊢ (𝑁‘𝐴) ∈ ℝ | ||
Theorem | nvs 26902 | Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) | ||
Theorem | nvsge0 26903 | The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = (𝐴 · (𝑁‘𝐵))) | ||
Theorem | nvm1 26904 | The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) | ||
Theorem | nvdif 26905 | The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(-1𝑆𝐵))) = (𝑁‘(𝐵𝐺(-1𝑆𝐴)))) | ||
Theorem | nvpi 26906 | The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(i𝑆𝐵))) = (𝑁‘(𝐵𝐺(-i𝑆𝐴)))) | ||
Theorem | nvz0 26907 | The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) = 0) | ||
Theorem | nvz 26908 | The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍)) | ||
Theorem | nvtri 26909 | Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
Theorem | nvmtri 26910 | Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
Theorem | nvabs 26911 | Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘((𝑁‘𝐴) − (𝑁‘𝐵))) ≤ (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) | ||
Theorem | nvge0 26912 | The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) | ||
Theorem | nvgt0 26913 | A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 𝑍 ↔ 0 < (𝑁‘𝐴))) | ||
Theorem | nv1 26914 | From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) | ||
Theorem | nvop 26915 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈〈𝐺, 𝑆〉, 𝑁〉) | ||
Theorem | cnnv 26916 | 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 | ||
Theorem | cnnvg 26917 | 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〉 ⇒ ⊢ + = ( +𝑣 ‘𝑈) | ||
Theorem | cnnvba 26918 | 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) | ||
Theorem | cnnvs 26919 | 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〉 ⇒ ⊢ · = ( ·𝑠OLD ‘𝑈) | ||
Theorem | cnnvnm 26920 | 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 = (normCV‘𝑈) | ||
Theorem | cnnvm 26921 | 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〉 ⇒ ⊢ − = ( −𝑣 ‘𝑈) | ||
Theorem | elimnv 26922 | 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 = (0vec‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 | ||
Theorem | elimnvu 26923 | 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 | ||
Theorem | imsval 26924 | 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 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) | ||
Theorem | imsdval 26925 | 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 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) | ||
Theorem | imsdval2 26926 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) | ||
Theorem | nvnd 26927 | 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 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) | ||
Theorem | imsdf 26928 | 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) | ||
Theorem | imsmetlem 26929 | Lemma for imsmet 26930. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = (inv‘𝐺) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
Theorem | imsmet 26930 | 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)) | ||
Theorem | imsxmet 26931 | 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)) | ||
Theorem | cnims 26932 | The metric induced on the complex numbers. cnmet 22385 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) | ||
Theorem | vacn 26933 | 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 = ( +𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | ||
Theorem | nmcvcn 26934 | 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 = (normCV‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (topGen‘ran (,)) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | nmcnc 26935 | The norm of a normed complex vector space is a continuous function to ℂ. (For ℝ, see nmcvcn 26934.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.) |
⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | smcnlem 26936* | Lemma for smcn 26937. (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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑇 = (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) ⇒ ⊢ 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽) | ||
Theorem | smcn 26937 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) | ||
Theorem | vmcn 26938 | 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 = ( −𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | ||
Syntax | cdip 26939 | Extend class notation with the class inner product functions. |
class ·𝑖OLD | ||
Definition | df-dip 26940* | 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 (1st ‘𝑤), the scalar product is (2nd ‘𝑤), 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) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4))) | ||
Theorem | dipfval 26941* | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))) | ||
Theorem | ipval 26942* | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) | ||
Theorem | ipval2lem2 26943 | Lemma for ipval3 26948. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℝ) | ||
Theorem | ipval2lem3 26944 | Lemma for ipval3 26948. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ) | ||
Theorem | ipval2lem4 26945 | Lemma for ipval3 26948. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℂ) | ||
Theorem | ipval2 26946 | Expansion of the inner product value ipval 26942. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4)) | ||
Theorem | 4ipval2 26947 | Four times the inner product value ipval3 26948, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴𝑃𝐵)) = ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))) | ||
Theorem | ipval3 26948 | Expansion of the inner product value ipval 26942. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝑀𝐵))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝑀(i𝑆𝐵)))↑2)))) / 4)) | ||
Theorem | ipidsq 26949 | The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2)) | ||
Theorem | ipnm 26950 | Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (√‘(𝐴𝑃𝐴))) | ||
Theorem | dipcl 26951 | An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) | ||
Theorem | ipf 26952 | Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃:(𝑋 × 𝑋)⟶ℂ) | ||
Theorem | dipcj 26953 | The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴)) | ||
Theorem | ipipcj 26954 | An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) · (𝐵𝑃𝐴)) = ((abs‘(𝐴𝑃𝐵))↑2)) | ||
Theorem | diporthcom 26955 | Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0)) | ||
Theorem | dip0r 26956 | Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝑍) = 0) | ||
Theorem | dip0l 26957 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝑃𝐴) = 0) | ||
Theorem | ipz 26958 | The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍)) | ||
Theorem | dipcn 26959 | Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) | ||
Syntax | css 26960 | Extend class notation with the class of all subspaces of normed complex vector spaces. |
class SubSp | ||
Definition | df-ssp 26961* | Define the class of all subspaces of complex normed vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))}) | ||
Theorem | sspval 26962* | The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}) | ||
Theorem | isssp 26963 | The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐹 = ( +𝑣 ‘𝑊) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) | ||
Theorem | sspid 26964 | A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) | ||
Theorem | sspnv 26965 | A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) | ||
Theorem | sspba 26966 | The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋) | ||
Theorem | sspg 26967 | Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐹 = ( +𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) | ||
Theorem | sspgval 26968 | Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐹 = ( +𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) | ||
Theorem | ssps 26969 | Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌))) | ||
Theorem | sspsval 26970 | Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵)) | ||
Theorem | sspmlem 26971* | Lemma for sspm 26973 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) & ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) & ⊢ (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑅) & ⊢ (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶𝑆) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) | ||
Theorem | sspmval 26972 | Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝐿 = ( −𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐿𝐵) = (𝐴𝑀𝐵)) | ||
Theorem | sspm 26973 | Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝐿 = ( −𝑣 ‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐿 = (𝑀 ↾ (𝑌 × 𝑌))) | ||
Theorem | sspz 26974 | The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑄 = (0vec‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑄 = 𝑍) | ||
Theorem | sspn 26975 | The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌)) | ||
Theorem | sspnval 26976 | The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ∧ 𝐴 ∈ 𝑌) → (𝑀‘𝐴) = (𝑁‘𝐴)) | ||
Theorem | sspimsval 26977 | The induced metric on a subspace in terms of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐶𝐵) = (𝐴𝐷𝐵)) | ||
Theorem | sspims 26978 | The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))) | ||
Syntax | clno 26979 | Extend class notation with the class of linear operators on normed complex vector spaces. |
class LnOp | ||
Syntax | cnmoo 26980 | Extend class notation with the class of operator norms on normed complex vector spaces. |
class normOpOLD | ||
Syntax | cblo 26981 | Extend class notation with the class of bounded linear operators on normed complex vector spaces. |
class BLnOp | ||
Syntax | c0o 26982 | Extend class notation with the class of zero operators on normed complex vector spaces. |
class 0op | ||
Definition | df-lno 26983* | Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e. the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
⊢ LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))}) | ||
Definition | df-nmoo 26984* | Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces 〈𝑢, 𝑤〉. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ))) | ||
Definition | df-blo 26985* | Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}) | ||
Definition | df-0o 26986* | Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | ||
Syntax | caj 26987 | Adjoint of an operator. |
class adj | ||
Syntax | chmo 26988 | Set of Hermitional (self-adjoint) operators. |
class HmOp | ||
Definition | df-aj 26989* | Define the adjoint of an operator (if it exists). The domain of 𝑈adj𝑊 is the set of all operators from 𝑈 to 𝑊 that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that 𝑈 and 𝑊 be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))}) | ||
Definition | df-hmo 26990* | Define the set of Hermitian (self-adjoint) operators on a normed complex vector space (normally a Hilbert space). Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) | ||
Theorem | lnoval 26991* | The set of linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐻 = ( +𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))}) | ||
Theorem | islno 26992* | The predicate "is a linear operator." (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐻 = ( +𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))) | ||
Theorem | lnolin 26993 | Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐻 = ( +𝑣 ‘𝑊) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶))) | ||
Theorem | lnof 26994 | A linear operator is a mapping. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌) | ||
Theorem | lno0 26995 | The value of a linear operator at zero is zero. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 18-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑄 = (0vec‘𝑈) & ⊢ 𝑍 = (0vec‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍) | ||
Theorem | lnocoi 26996 | The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑀 = (𝑊 LnOp 𝑋) & ⊢ 𝑁 = (𝑈 LnOp 𝑋) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑋 ∈ NrmCVec & ⊢ 𝑆 ∈ 𝐿 & ⊢ 𝑇 ∈ 𝑀 ⇒ ⊢ (𝑇 ∘ 𝑆) ∈ 𝑁 | ||
Theorem | lnoadd 26997 | Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐻 = ( +𝑣 ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝐺𝐵)) = ((𝑇‘𝐴)𝐻(𝑇‘𝐵))) | ||
Theorem | lnosub 26998 | Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = ( −𝑣 ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑀𝐵)) = ((𝑇‘𝐴)𝑁(𝑇‘𝐵))) | ||
Theorem | lnomul 26999 | Scalar multiplication property of a linear operator. (Contributed by NM, 5-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋)) → (𝑇‘(𝐴𝑅𝐵)) = (𝐴𝑆(𝑇‘𝐵))) | ||
Theorem | nvo00 27000 | Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍})) |
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