P. 292 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ipasslem9 29101	Lemma for ipassi 29104. Conclude from ipasslem8 29100 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 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    ⇒   ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))
Theorem	ipasslem10 29102	Lemma for ipassi 29104. 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 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))
Theorem	ipasslem11 29103	Lemma for ipassi 29104. 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 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    ⇒   ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))
Theorem	ipassi 29104	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 ∈ CPreHilOLD    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))
Theorem	dipdir 29105	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 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))
Theorem	dipdi 29106	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 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶)))
Theorem	ip2dii 29107	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 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    &   ⊢ 𝐶 ∈ 𝑋    &   ⊢ 𝐷 ∈ 𝑋    ⇒   ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶)))
Theorem	dipass 29108	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 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))
Theorem	dipassr 29109	"Associative" law for second argument of inner product (compare dipass 29108). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶)))
Theorem	dipassr2 29110	"Associative" law for inner product. Conjugate version of dipassr 29109. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶)))
Theorem	dipsubdir 29111	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 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶)))
Theorem	dipsubdi 29112	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 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶)))
Theorem	pythi 29113	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 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    ⇒   ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
Theorem	siilem1 29114	Lemma for sii 29117. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    &   ⊢ 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 29115	Lemma for sii 29117. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    &   ⊢ 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 29116	Inference from sii 29117. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    &   ⊢ 𝐴 ∈ 𝑋    &   ⊢ 𝐵 ∈ 𝑋    ⇒   ⊢ (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))
Theorem	sii 29117	Obsolete version of ipcau 24307 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 29383, bcsiALT 29442, bcsiHIL 29443, csbren 24468. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    ⇒   ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))
Theorem	ipblnfi 29118*	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 ∈ CPreHilOLD    &   ⊢ 𝐶 = ⟨⟨ + , · ⟩, abs⟩    &   ⊢ 𝐵 = (𝑈 BLnOp 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑥𝑃𝐴))    ⇒   ⊢ (𝐴 ∈ 𝑋 → 𝐹 ∈ 𝐵)
Theorem	ip2eqi 29119*	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 ∈ CPreHilOLD    ⇒   ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵))
Theorem	phoeqi 29120*	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 ∈ CPreHilOLD    ⇒   ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇))
Theorem	ajmoi 29121*	Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ CPreHilOLD    ⇒   ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))
Theorem	ajfuni 29122	The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
⊢ 𝐴 = (𝑈adj𝑊)    &   ⊢ 𝑈 ∈ CPreHilOLD    &   ⊢ 𝑊 ∈ NrmCVec    ⇒   ⊢ Fun 𝐴
Theorem	ajfun 29123	The adjoint function is a function. This is not immediately apparent from df-aj 29013 but results from the uniqueness shown by ajmoi 29121. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
⊢ 𝐴 = (𝑈adj𝑊)    ⇒   ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴)
Theorem	ajval 29124*	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 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
18.6  Complex Banach spaces
18.6.1  Definition and basic properties
Syntax	ccbn 29125	Extend class notation with the class of all complex Banach spaces.
class CBan
Definition	df-cbn 29126	Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 24405 instead. (New usage is discouraged.)
⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}
Theorem	iscbn 29127	A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24407 instead. (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))
Theorem	cbncms 29128	The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24416 (or preferably bncms 24413) instead. (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
Theorem	bnnv 29129	Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 24409 instead. (New usage is discouraged.)
⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
Theorem	bnrel 29130	The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CBan
Theorem	bnsscmcl 29131	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)))
18.6.2  Examples of complex Banach spaces
Theorem	cnbn 29132	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
18.6.3  Uniform Boundedness Theorem
Theorem	ubthlem1 29133*	Lemma for ubth 29136. 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 24399, 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 = (normCV‘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 29134*	Lemma for ubth 29136. 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 = (normCV‘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 normOpOLD 𝑊)‘𝑡) ≤ 𝑑)
Theorem	ubthlem3 29135*	Lemma for ubth 29136. Prove the reverse implication, using nmblolbi 29063. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘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 normOpOLD 𝑊)‘𝑡) ≤ 𝑑))
Theorem	ubth 29136*	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 = (normCV‘𝑊)    &   ⊢ 𝑀 = (𝑈 normOpOLD 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))
18.6.4  Minimizing Vector Theorem
Theorem	minvecolem1 29137*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29138*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29139*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29140*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29141*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29142*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29143*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29144*	Lemma for minveco 29147. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10122. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29145*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29146*	Lemma for minveco 29147. 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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (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 29147*	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 = (normCV‘𝑈)    &   ⊢ 𝑌 = (BaseSet‘𝑊)    &   ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)    &   ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))
18.7  Complex Hilbert spaces
18.7.1  Definition and basic properties
Syntax	chlo 29148	Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD
Definition	df-hlo 29149	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.)
⊢ CHilOLD = (CBan ∩ CPreHilOLD)
Theorem	ishlo 29150	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 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
Theorem	hlobn 29151	Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)
Theorem	hlph 29152	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 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD)
Theorem	hlrel 29153	The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CHilOLD
Theorem	hlnv 29154	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
Theorem	hlnvi 29155	Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
⊢ 𝑈 ∈ CHilOLD    ⇒   ⊢ 𝑈 ∈ NrmCVec
Theorem	hlvc 29156	Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑊 = (1st ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑊 ∈ CVecOLD)
Theorem	hlcmet 29157	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 ∈ CHilOLD → 𝐷 ∈ (CMet‘𝑋))
Theorem	hlmet 29158	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 ∈ CHilOLD → 𝐷 ∈ (Met‘𝑋))
Theorem	hlpar2 29159	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 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
Theorem	hlpar 29160	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 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
18.7.2  Standard axioms for a complex Hilbert space
Theorem	hlex 29161	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 29162	Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Theorem	hlcom 29163	Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Theorem	hlass 29164	Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
Theorem	hl0cl 29165	The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑍 ∈ 𝑋)
Theorem	hladdid 29166	Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
Theorem	hlmulf 29167	Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
Theorem	hlmulid 29168	Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
Theorem	hlmulass 29169	Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
Theorem	hldi 29170	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 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
Theorem	hldir 29171	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 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Theorem	hlmul0 29172	Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)
Theorem	hlipf 29173	Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ (𝑈 ∈ CHilOLD → 𝑃:(𝑋 × 𝑋)⟶ℂ)
Theorem	hlipcj 29174	Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (∗‘(𝐵𝑃𝐴)))
Theorem	hlipdir 29175	Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))
Theorem	hlipass 29176	Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))
Theorem	hlipgt0 29177	The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → 0 < (𝐴𝑃𝐴))
Theorem	hlcompl 29178	Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)
⊢ 𝐷 = (IndMet‘𝑈)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽))
18.7.3  Examples of complex Hilbert spaces
Theorem	cnchl 29179	The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩    ⇒   ⊢ 𝑈 ∈ CHilOLD
18.7.4  Hellinger-Toeplitz Theorem
Theorem	htthlem 29180*	Lemma for htth 29181. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = ⟨𝑤 ∣ 𝑇(𝑧)⟩ = ⟨𝑇(𝑤) ∣ 𝑧⟩ for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 29118, so by the Uniform Boundedness theorem ubth 29136, there is a uniform bound 𝑦 on ∥ 𝐹(𝑥) ∥ for all 𝑥 in the unit ball. Then ∣ 𝑇(𝑥) ∣ ↑2 = ⟨𝑇(𝑥) ∣ 𝑇(𝑥)⟩ = 𝐹(𝑥)( 𝑇(𝑥)) ≤ 𝑦 ∣ 𝑇(𝑥) ∣, so ∣ 𝑇(𝑥) ∣ ≤ 𝑦 and 𝑇 is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑈)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑈 ∈ CHilOLD    &   ⊢ 𝑊 = ⟨⟨ + , · ⟩, abs⟩    &   ⊢ (𝜑 → 𝑇 ∈ 𝐿)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦))    &   ⊢ 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝑤 ∈ 𝑋 ↦ (𝑤𝑃(𝑇‘𝑧))))    &   ⊢ 𝐾 = (𝐹 “ {𝑧 ∈ 𝑋 ∣ (𝑁‘𝑧) ≤ 1})    ⇒   ⊢ (𝜑 → 𝑇 ∈ 𝐵)
Theorem	htth 29181*	Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑃 = (·𝑖OLD‘𝑈)    &   ⊢ 𝐿 = (𝑈 LnOp 𝑈)    &   ⊢ 𝐵 = (𝑈 BLnOp 𝑈)    ⇒   ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵)
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED) This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should instead work with general Hilbert spaces as defined by df-hil 20821; note that df-hil 20821 uses extensible structures. The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details.
19.1  Axiomatization of complex pre-Hilbert spaces
19.1.1  Basic Hilbert space definitions
Syntax	chba 29182	Extend class notation with Hilbert vector space.
class ℋ
Syntax	cva 29183	Extend class notation with vector addition in Hilbert space. In the literature, the subscript "h" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 10805.
class +ℎ
Syntax	csm 29184	Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class ·ℎ
Syntax	csp 29185	Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩ but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4565.
class ·ih
Syntax	cno 29186	Extend class notation with the norm function in Hilbert space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions.
class normℎ
Syntax	c0v 29187	Extend class notation with zero vector in Hilbert space.
class 0ℎ
Syntax	cmv 29188	Extend class notation with vector subtraction in Hilbert space.
class −ℎ
Syntax	ccauold 29189	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 29190	Extend class notation with convergence relation in Hilbert space.
class ⇝𝑣
Syntax	csh 29191	Extend class notation with set of subspaces of a Hilbert space.
class Sℋ
Syntax	cch 29192	Extend class notation with set of closed subspaces of a Hilbert space.
class Cℋ
Syntax	cort 29193	Extend class notation with orthogonal complement in Cℋ.
class ⊥
Syntax	cph 29194	Extend class notation with subspace sum in Cℋ.
class +ℋ
Syntax	cspn 29195	Extend class notation with subspace span in Cℋ.
class span
Syntax	chj 29196	Extend class notation with join in Cℋ.
class ∨ℋ
Syntax	chsup 29197	Extend class notation with supremum of a collection in Cℋ.
class ∨ℋ
Syntax	c0h 29198	Extend class notation with zero of Cℋ.
class 0ℋ
Syntax	ccm 29199	Extend class notation with the commutes relation on a Hilbert lattice.
class 𝐶ℋ
Syntax	cpjh 29200	Extend class notation with set of projections on a Hilbert space.
class projℎ