P. 302 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 30101-30200 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	hmoval 30101*	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 30102	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 30103	Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHil_OLD

Definition	df-ph 30104*	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 30105	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 30106	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 30107	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 30108*	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 30109	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 30110	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 30111	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 30112	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 30113*	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 30114	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 30115	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 30116	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 30117	Lemma for ip1i 30118. (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 30118	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 30119	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 30120	Lemma for ipdiri 30121. (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 30121	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 30122	Lemma for ipassi 30132. 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 30123	Lemma for ipassi 30132. 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 30124	Lemma for ipassi 30132. 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 30125	Lemma for ipassi 30132. 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 30126	Lemma for ipassi 30132. 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 30127*	Lemma for ipassi 30132. 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 30128*	Lemma for ipassi 30132. By ipasslem5 30126, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 24320, 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 30129	Lemma for ipassi 30132. Conclude from ipasslem8 30128 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 30130	Lemma for ipassi 30132. 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 30131	Lemma for ipassi 30132. 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 30132	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 30133	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)))

Theorem	dipdi 30134	Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶)))

Theorem	ip2dii 30135	Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝐷 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶)))

Theorem	dipass 30136	Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theorem	dipassr 30137	"Associative" law for second argument of inner product (compare dipass 30136). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶)))

Theorem	dipassr2 30138	"Associative" law for inner product. Conjugate version of dipassr 30137. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶)))

Theorem	dipsubdir 30139	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶)))

Theorem	dipsubdi 30140	Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶)))

Theorem	pythi 30141	The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space 𝑈. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))

Theorem	siilem1 30142	Lemma for sii 30145. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝐶 ∈ ℂ & ⊢ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ & ⊢ 0 ≤ (𝐶 · (𝐴𝑃𝐵)) ⇒ ⊢ ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))

Theorem	siilem2 30143	Lemma for sii 30145. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))))

Theorem	siii 30144	Inference from sii 30145. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))

Theorem	sii 30145	Obsolete version of ipcau 24762 as of 22-Sep-2024. Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also Theorems bcseqi 30411, bcsiALT 30470, bcsiHIL 30471, csbren 24923. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))

Theorem	ipblnfi 30146*	A function 𝐹 generated by varying the first argument of an inner product (with its second argument a fixed vector 𝐴) is a bounded linear functional, i.e. a bounded linear operator from the vector space to ℂ. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝐶 = ⟨⟨ + , · ⟩, abs⟩ & ⊢ 𝐵 = (𝑈 BLnOp 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑥𝑃𝐴)) ⇒ ⊢ (𝐴 ∈ 𝑋 → 𝐹 ∈ 𝐵)

Theorem	ip2eqi 30147*	Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵))

Theorem	phoeqi 30148*	A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇))

Theorem	ajmoi 30149*	Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHil_OLD ⇒ ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))

Theorem	ajfuni 30150	The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝐴 = (𝑈adj𝑊) & ⊢ 𝑈 ∈ CPreHil_OLD & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ Fun 𝐴

Theorem	ajfun 30151	The adjoint function is a function. This is not immediately apparent from df-aj 30041 but results from the uniqueness shown by ajmoi 30149. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴)

Theorem	ajval 30152*	Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·_𝑖OLD‘𝑈) & ⊢ 𝑄 = (·_𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))

19.6 Complex Banach spaces

19.6.1 Definition and basic properties

Syntax	ccbn 30153	Extend class notation with the class of all complex Banach spaces.
class CBan

Definition	df-cbn 30154	Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 24860 instead. (New usage is discouraged.)
⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}

Theorem	iscbn 30155	A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24862 instead. (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Theorem	cbncms 30156	The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24871 (or preferably bncms 24868) instead. (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Theorem	bnnv 30157	Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 24864 instead. (New usage is discouraged.)
⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Theorem	bnrel 30158	The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CBan

Theorem	bnsscmcl 30159	A subspace of a Banach space is a Banach space iff it is closed in the norm-induced metric of the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐻 = (SubSp‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) ⇒ ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ 𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽)))

19.6.2 Examples of complex Banach spaces

Theorem	cnbn 30160	The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩ ⇒ ⊢ 𝑈 ∈ CBan

19.6.3 Uniform Boundedness Theorem

Theorem	ubthlem1 30161*	Lemma for ubth 30164. The function 𝐴 exhibits a countable collection of sets that are closed, being the inverse image under 𝑡 of the closed ball of radius 𝑘, and by assumption they cover 𝑋. Thus, by the Baire Category theorem bcth2 24854, for some 𝑛 the set 𝐴‘𝑛 has an interior, meaning that there is a closed ball {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐) & ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))

Theorem	ubthlem2 30162*	Lemma for ubth 30164. Given that there is a closed ball 𝐵(𝑃, 𝑅) in 𝐴‘𝐾, for any 𝑥 ∈ 𝐵(0, 1), we have 𝑃 + 𝑅 · 𝑥 ∈ 𝐵(𝑃, 𝑅) and 𝑃 ∈ 𝐵(𝑃, 𝑅), so both of these have norm(𝑡(𝑧)) ≤ 𝐾 and so norm(𝑡(𝑥 )) ≤ (norm(𝑡(𝑃)) + norm(𝑡(𝑃 + 𝑅 · 𝑥))) / 𝑅 ≤ ( 𝐾 + 𝐾) / 𝑅, which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐) & ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ⁺) & ⊢ (𝜑 → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴‘𝐾)) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑)

Theorem	ubthlem3 30163*	Lemma for ubth 30164. Prove the reverse implication, using nmblolbi 30091. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CBan & ⊢ 𝑊 ∈ NrmCVec & ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 ((𝑈 normOp_OLD 𝑊)‘𝑡) ≤ 𝑑))

Theorem	ubth 30164*	Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let 𝑇 be a collection of bounded linear operators on a Banach space. If, for every vector 𝑥, the norms of the operators' values are bounded, then the operators' norms are also bounded. Theorem 4.7-3 of [Kreyszig] p. 249. See also http://en.wikipedia.org/wiki/Uniform_boundedness_principle. (Contributed by NM, 7-Nov-2007.) (Proof shortened by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑊) & ⊢ 𝑀 = (𝑈 normOp_OLD 𝑊) ⇒ ⊢ ((𝑈 ∈ CBan ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ⊆ (𝑈 BLnOp 𝑊)) → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑀‘𝑡) ≤ 𝑑))

19.6.4 Minimizing Vector Theorem

Theorem	minvecolem1 30165*	Lemma for minveco 30175. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⇒ ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Theorem	minvecolem2 30166*	Lemma for minveco 30175. Any two points 𝐾 and 𝐿 in 𝑌 are close to each other if they are close to the infimum of distance to 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ 𝑌) & ⊢ (𝜑 → 𝐿 ∈ 𝑌) & ⊢ (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵)) & ⊢ (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵)) ⇒ ⊢ (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))

Theorem	minvecolem3 30167*	Lemma for minveco 30175. The sequence formed by taking elements successively closer to the infimum is Cauchy. (Contributed by Mario Carneiro, 8-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))

Theorem	minvecolem4a 30168*	Lemma for minveco 30175. 𝐹 is convergent in the subspace topology on 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝐹(⇝_𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝_𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))

Theorem	minvecolem4b 30169*	Lemma for minveco 30175. The convergent point of the cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → ((⇝_𝑡‘𝐽)‘𝐹) ∈ 𝑋)

Theorem	minvecolem4c 30170*	Lemma for minveco 30175. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) ⇒ ⊢ (𝜑 → 𝑆 ∈ ℝ)

Theorem	minvecolem4 30171*	Lemma for minveco 30175. The convergent point of the cauchy sequence 𝐹 attains the minimum distance, and so is closer to 𝐴 than any other point in 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) & ⊢ 𝑇 = (1 / (((((𝐴𝐷((⇝_𝑡‘𝐽)‘𝐹)) + 𝑆) / 2)↑2) − (𝑆↑2))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theorem	minvecolem5 30172*	Lemma for minveco 30175. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10432. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theorem	minvecolem6 30173*	Lemma for minveco 30175. Any minimal point is less than 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))))

Theorem	minvecolem7 30174*	Lemma for minveco 30175. Since any two minimal points are distance zero away from each other, the minimal point is unique. (Contributed by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) & ⊢ 𝑆 = inf(𝑅, ℝ, < ) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theorem	minveco 30175*	Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD) & ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

19.7 Complex Hilbert spaces

19.7.1 Definition and basic properties

Syntax	chlo 30176	Extend class notation with the class of all complex Hilbert spaces.
class CHil_OLD

Definition	df-hlo 30177	Define the class of all complex Hilbert spaces. A Hilbert space is a Banach space which is also an inner product space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ CHil_OLD = (CBan ∩ CPreHil_OLD)

Theorem	ishlo 30178	The predicate "is a complex Hilbert space." A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHil_OLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHil_OLD))

Theorem	hlobn 30179	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ CBan)

Theorem	hlph 30180	Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ CPreHil_OLD)

Theorem	hlrel 30181	The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CHil_OLD

Theorem	hlnv 30182	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ NrmCVec)

Theorem	hlnvi 30183	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
⊢ 𝑈 ∈ CHil_OLD ⇒ ⊢ 𝑈 ∈ NrmCVec

Theorem	hlvc 30184	Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑊 = (1^st ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝑊 ∈ CVec_OLD)

Theorem	hlcmet 30185	The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝐷 ∈ (CMet‘𝑋))

Theorem	hlmet 30186	The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝐷 ∈ (Met‘𝑋))

Theorem	hlpar2 30187	The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑀 = ( −_𝑣 ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Theorem	hlpar 30188	The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑁 = (norm_CV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

19.7.2 Standard axioms for a complex Hilbert space

Theorem	hlex 30189	The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ 𝑋 ∈ V

Theorem	hladdf 30190	Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Theorem	hlcom 30191	Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theorem	hlass 30192	Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theorem	hl0cl 30193	The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝑍 ∈ 𝑋)

Theorem	hladdid 30194	Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theorem	hlmulf 30195	Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ (𝑈 ∈ CHil_OLD → 𝑆:(ℂ × 𝑋)⟶𝑋)

Theorem	hlmulid 30196	Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Theorem	hlmulass 30197	Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theorem	hldi 30198	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theorem	hldir 30199	Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +_𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theorem	hlmul0 30200	Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0_vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)

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