P. 309 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 30801-30900 *Has distinct variable group(s)
Type	Label	Description
Statement

Syntax	c0o 30801	Extend class notation with the class of zero operators on normed complex vector spaces.
class 0_op

Definition	df-lno 30802*	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 30803*	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.)
⊢ normOp_OLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑_m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((norm_CV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((norm_CV‘𝑤)‘(𝑡‘𝑧)))}, ℝ^*, < )))

Definition	df-blo 30804*	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 normOp_OLD 𝑤)‘𝑡) < +∞})

Definition	df-0o 30805*	Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 0_op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0_vec‘𝑤)}))

Syntax	caj 30806	Adjoint of an operator.
class adj

Syntax	chmo 30807	Set of Hermitional (self-adjoint) operators.
class HmOp

Definition	df-aj 30808*	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 30809*	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 30810*	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 30811*	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 30812	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 30813	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 30814	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 = (0_vec‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑄) = 𝑍)

Theorem	lnocoi 30815	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 30816	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 30817	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 30818	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 30819	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 30820*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑_m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ^*, < )))

Theorem	nmooval 30821*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))

Theorem	nmosetre 30822*	The set in the supremum of the operator norm definition df-nmoo 30803 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (norm_CV‘𝑊) ⇒ ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ)

Theorem	nmosetn0 30823*	The set in the supremum of the operator norm definition df-nmoo 30803 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))})

Theorem	nmoxr 30824	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 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ^*)

Theorem	nmooge0 30825	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 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇))

Theorem	nmorepnf 30826	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 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))

Theorem	nmoreltpnf 30827	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 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞))

Theorem	nmogtmnf 30828	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 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇))

Theorem	nmoolb 30829	A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇))

Theorem	nmoubi 30830*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ^*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴)))

Theorem	nmoub3i 30831*	An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴))

Theorem	nmoub2i 30832*	An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30833*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))

Theorem	nmounbi 30834*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))

Theorem	nmounbseqi 30835*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30836*	Alternate shorter proof of nmounbseqi 30835 based on Axioms ax-reg 9501 and ax-ac2 10377 instead of ax-cc 10349. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30837*	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30838*	Alternate shorter proof of nmobndseqi 30837 based on Axioms ax-reg 9501 and ax-ac2 10377 instead of ax-cc 10349. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30839*	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 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞})

Theorem	isblo 30840	The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞)))

Theorem	isblo2 30841	The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ)))

Theorem	bloln 30842	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 30843	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 30844	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 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ)

Theorem	0ofval 30845	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 = (0_vec‘𝑊) & ⊢ 𝑂 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Theorem	0oval 30846	Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑊) & ⊢ 𝑂 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍)

Theorem	0oo 30847	The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌)

Theorem	0lno 30848	The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿)

Theorem	nmoo0 30849	The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0)

Theorem	0blo 30850	The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐵)

Theorem	nmlno0lem 30851	Lemma for nmlno0i 30852. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊) & ⊢ 𝑃 = (0_vec‘𝑈) & ⊢ 𝑄 = (0_vec‘𝑊) & ⊢ 𝐾 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) ⇒ ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)

Theorem	nmlno0i 30852	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 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))

Theorem	nmlno0 30853	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 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍))

Theorem	nmlnoubi 30854*	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 = (0_vec‘𝑈) & ⊢ 𝐾 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 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 30855	The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.)
⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝑍 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇)))

Theorem	lnon0 30856*	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 = (0_vec‘𝑈) & ⊢ 𝑂 = (𝑈 0_op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍)

Theorem	nmblolbii 30857	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Theorem	nmblolbi 30858	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 = (norm_CV‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑊) & ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴)))

Theorem	isblo3i 30859*	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 = (norm_CV‘𝑈) & ⊢ 𝑁 = (norm_CV‘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 30860*	Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (norm_CV‘𝑈) & ⊢ 𝑁 = (norm_CV‘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 30861	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 normOp_OLD 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 30862	Lemma for blocni 30863 and lnocni 30864. 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 30863	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 30864	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 30865	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 30866	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 30867*	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 30868*	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 30869	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)))

19.5 Inner product (pre-Hilbert) spaces

19.5.1 Definition and basic properties

Syntax	ccphlo 30870	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil_OLD

Definition	df-ph 30871*	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.)
⊢ CPreHil_OLD = (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 30872	Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)

Theorem	phrel 30873	The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
⊢ Rel CPreHil_OLD

Theorem	phnvi 30874	Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ 𝑈 ∈ NrmCVec

Theorem	isphg 30875*	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⟩ ∈ CPreHil_OLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Theorem	phop 30876	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 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

19.5.2 Examples of pre-Hilbert spaces

Theorem	cncph 30877	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 ∈ CPreHil_OLD

Theorem	elimph 30878	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 = (0_vec‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋

Theorem	elimphu 30879	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 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHil_OLD

19.5.3 Properties of pre-Hilbert spaces

Theorem	isph 30880*	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 = (norm_CV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))

Theorem	phpar2 30881	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 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Theorem	phpar 30882	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 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Theorem	ip0i 30883	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 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (norm_CV‘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 30884	Lemma for ip1i 30885. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theorem	ip1i 30885	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 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theorem	ip2i 30886	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 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵))

Theorem	ipdirilem 30887	Lemma for ipdiri 30888. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

Theorem	ipdiri 30888	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 ∈ CPreHil_OLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theorem	ipasslem1 30889	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theorem	ipasslem2 30890	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵)))

Theorem	ipasslem3 30891	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theorem	ipasslem4 30892	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵)))

Theorem	ipasslem5 30893	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐶 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theorem	ipasslem7 30894*	Lemma for ipassi 30899. 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 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂ_fld) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn 𝐾)

Theorem	ipasslem8 30895*	Lemma for ipassi 30899. By ipasslem5 30893, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24745, 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 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ⇒ ⊢ 𝐹:ℝ⟶{0}

Theorem	ipasslem9 30896	Lemma for ipassi 30899. Conclude from ipasslem8 30895 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theorem	ipasslem10 30897	Lemma for ipassi 30899. Show the inner product associative law for the imaginary number i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

Theorem	ipasslem11 30898	Lemma for ipassi 30899. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theorem	ipassi 30899	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theorem	dipdir 30900	Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49400 495 49401-49500 496 49501-49600 497 49601-49700 498 49701-49800 499 49801-49900 500 49901-50000 501 50001-50086

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