HomeTheorem List (p. 312 of 507)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | sspimsval 31101 | 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 31102 | 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 31103 | Extend class notation with the class of linear operators on normed complex vector spaces. |
| class LnOp | ||
| Syntax | cnmoo 31104 | Extend class notation with the class of operator norms on normed complex vector spaces. |
| class normOpOLD | ||
| Syntax | cblo 31105 | Extend class notation with the class of bounded linear operators on normed complex vector spaces. |
| class BLnOp | ||
| Syntax | c0o 31106 | Extend class notation with the class of zero operators on normed complex vector spaces. |
| class 0op | ||
| Definition | df-lno 31107* | 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 31108* | 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 31109* | 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 31110* | 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 31111 | Adjoint of an operator. |
| class adj | ||
| Syntax | chmo 31112 | Set of Hermitional (self-adjoint) operators. |
| class HmOp | ||
| Definition | df-aj 31113* | 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 31114* | 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 31115* | 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 31116* | 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 31117 | 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 31118 | 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 31119 | 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 31120 | 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 31121 | 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 31122 | 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 31123 | 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 31124 | Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍})) | ||
| Theorem | nmoofval 31125* | The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))) | ||
| Theorem | nmooval 31126* | The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ*, < )) | ||
| Theorem | nmosetre 31127* | The set in the supremum of the operator norm definition df-nmoo 31108 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (normCV‘𝑊) ⇒ ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ) | ||
| Theorem | nmosetn0 31128* | The set in the supremum of the operator norm definition df-nmoo 31108 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))}) | ||
| Theorem | nmoxr 31129 | The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*) | ||
| Theorem | nmooge0 31130 | The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇)) | ||
| Theorem | nmorepnf 31131 | The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞)) | ||
| Theorem | nmoreltpnf 31132 | The norm of any operator is real iff it is less than plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞)) | ||
| Theorem | nmogtmnf 31133 | The norm of an operator is greater than minus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇)) | ||
| Theorem | nmoolb 31134 | A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇)) | ||
| Theorem | nmoubi 31135* | An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
| Theorem | nmoub3i 31136* | An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴)) | ||
| Theorem | nmoub2i 31137* | An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ 𝐴) | ||
| Theorem | nmobndi 31138* | Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟))) | ||
| Theorem | nmounbi 31139* | Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))) | ||
| Theorem | nmounbseqi 31140* | An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘)))))) | ||
| Theorem | nmounbseqiALT 31141* | Alternate shorter proof of nmounbseqi 31140 based on Axioms ax-reg 9552 and ax-ac2 10453 instead of ax-cc 10425. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘)))))) | ||
| Theorem | nmobndseqi 31142* | A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | nmobndseqiALT 31143* | Alternate shorter proof of nmobndseqi 31142 based on Axioms ax-reg 9552 and ax-ac2 10453 instead of ax-cc 10425. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | bloval 31144* | The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) | ||
| Theorem | isblo 31145 | The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) | ||
| Theorem | isblo2 31146 | The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ))) | ||
| Theorem | bloln 31147 | A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿) | ||
| Theorem | blof 31148 | A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) | ||
| Theorem | nmblore 31149 | The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | 0ofval 31150 | The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑊) & ⊢ 𝑂 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) | ||
| Theorem | 0oval 31151 | Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑊) & ⊢ 𝑂 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍) | ||
| Theorem | 0oo 31152 | The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌) | ||
| Theorem | 0lno 31153 | The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿) | ||
| Theorem | nmoo0 31154 | The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0) | ||
| Theorem | 0blo 31155 | The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐵) | ||
| Theorem | nmlno0lem 31156 | Lemma for nmlno0i 31157. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝑃 = (0vec‘𝑈) & ⊢ 𝑄 = (0vec‘𝑊) & ⊢ 𝐾 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) ⇒ ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍) | ||
| Theorem | nmlno0i 31157 | The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlno0 31158 | The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlnoubi 31159* | An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝐾 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ≠ 𝑍 → (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐾‘𝑥)))) → (𝑁‘𝑇) ≤ 𝐴) | ||
| Theorem | nmlnogt0 31160 | The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇))) | ||
| Theorem | lnon0 31161* | The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑂 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) | ||
| Theorem | nmblolbii 31162 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | nmblolbi 31163 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | isblo3i 31164* | The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)))) | ||
| Theorem | blo3i 31165* | Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) | ||
| Theorem | blometi 31166 | Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) ≤ ((𝑁‘𝑇) · (𝑃𝐶𝑄))) | ||
| Theorem | blocnilem 31167 | Lemma for blocni 31168 and lnocni 31169. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵) | ||
| Theorem | blocni 31168 | A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 ⇒ ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) | ||
| Theorem | lnocni 31169 | If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | blocn 31170 | A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) | ||
| Theorem | blocn2 31171 | A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | ajfval 31172* | The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}) | ||
| Theorem | hmoval 31173* | The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) | ||
| Theorem | ishmo 31174 | The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) | ||
| Syntax | ccphlo 31175 | Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces). |
| class CPreHilOLD | ||
| Definition | df-ph 31176* | Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is 𝑔, the scalar product is 𝑠, and the norm is 𝑛. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) | ||
| Theorem | phnv 31177 | Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | ||
| Theorem | phrel 31178 | The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ Rel CPreHilOLD | ||
| Theorem | phnvi 31179 | Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ 𝑈 ∈ NrmCVec | ||
| Theorem | isphg 31180* | The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))) | ||
| Theorem | phop 31181 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩) | ||
| Theorem | cncph 31182 | The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩ ⇒ ⊢ 𝑈 ∈ CPreHilOLD | ||
| Theorem | elimph 31183 | Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 | ||
| Theorem | elimphu 31184 | Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.) |
| ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHilOLD | ||
| Theorem | isph 31185* | The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))) | ||
| Theorem | phpar2 31186 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | phpar 31187 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | ip0i 31188 | A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where 𝐽 is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))) | ||
| Theorem | ip1ilem 31189 | Lemma for ip1i 31190. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip1i 31190 | Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip2i 31191 | Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) | ||
| Theorem | ipdirilem 31192 | Lemma for ipdiri 31193. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) | ||
| Theorem | ipdiri 31193 | Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
| Theorem | ipasslem1 31194 | Lemma for ipassi 31204. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem2 31195 | Lemma for ipassi 31204. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem3 31196 | Lemma for ipassi 31204. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem4 31197 | Lemma for ipassi 31204. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem5 31198 | Lemma for ipassi 31204. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐶 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem7 31199* | Lemma for ipassi 31204. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on ℝ. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) | ||
| Theorem | ipasslem8 31200* | Lemma for ipassi 31204. By ipasslem5 31198, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24965, we conclude 𝐹 is 0 for all ℝ. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ⇒ ⊢ 𝐹:ℝ⟶{0} | ||
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