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Theorem List for Metamath Proof Explorer - 28601-28700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremip0i 28601 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   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)))

Theoremip1ilem 28602 Lemma for ip1i 28603. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   𝐽 ∈ ℂ       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip1i 28603 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip2i 28604 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵))

Theoremipdirilem 28605 Lemma for ipdiri 28606. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

Theoremipdiri 28606 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremipasslem1 28607 Lemma for ipassi 28617. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem2 28608 Lemma for ipassi 28617. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem3 28609 Lemma for ipassi 28617. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℤ ∧ 𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem4 28610 Lemma for ipassi 28617. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ ∧ 𝐴𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵)))

Theoremipasslem5 28611 Lemma for ipassi 28617. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝐶 ∈ ℚ ∧ 𝐴𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem7 28612* Lemma for ipassi 28617. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ (𝐽 Cn 𝐾)

Theoremipasslem8 28613* Lemma for ipassi 28617. By ipasslem5 28611, 𝐹 is 0 for all ; since it is continuous and is dense in by qdensere2 23404, we conclude 𝐹 is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))       𝐹:ℝ⟶{0}

Theoremipasslem9 28614 Lemma for ipassi 28617. Conclude from ipasslem8 28613 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem10 28615 Lemma for ipassi 28617. 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑁 = (normCV𝑈)       ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

Theoremipasslem11 28616 Lemma for ipassi 28617. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipassi 28617 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipdir 28618 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 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremdipdi 28619 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶)))

Theoremip2dii 28620 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝐷𝑋       ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶)))

Theoremdipass 28621 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 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipassr 28622 "Associative" law for second argument of inner product (compare dipass 28621). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶)))

Theoremdipassr2 28623 "Associative" law for inner product. Conjugate version of dipassr 28622. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶)))

Theoremdipsubdir 28624 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶)))

Theoremdipsubdi 28625 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶)))

Theorempythi 28626 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. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))

Theoremsiilem1 28627 Lemma for sii 28630. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝐶 ∈ ℂ    &   (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ    &   0 ≤ (𝐶 · (𝐴𝑃𝐵))       ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵)))

Theoremsiilem2 28628 Lemma for sii 28630. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵))))

Theoremsiii 28629 Inference from sii 28630. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵))

Theoremsii 28630 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 28896, bcsiALT 28955, bcsiHIL 28956, csbren 24001. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵)))

Theoremipblnfi 28631* 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 . (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐶 = ⟨⟨ + , · ⟩, abs⟩    &   𝐵 = (𝑈 BLnOp 𝐶)    &   𝐹 = (𝑥𝑋 ↦ (𝑥𝑃𝐴))       (𝐴𝑋𝐹𝐵)

Theoremip2eqi 28632* 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 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵))

Theoremphoeqi 28633* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝑆:𝑌𝑋𝑇:𝑌𝑋) → (∀𝑥𝑋𝑦𝑌 (𝑥𝑃(𝑆𝑦)) = (𝑥𝑃(𝑇𝑦)) ↔ 𝑆 = 𝑇))

Theoremajmoi 28634* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ∃*𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))

Theoremajfuni 28635 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)    &   𝑈 ∈ CPreHilOLD    &   𝑊 ∈ NrmCVec       Fun 𝐴

Theoremajfun 28636 The adjoint function is a function. This is not immediately apparent from df-aj 28526 but results from the uniqueness shown by ajmoi 28634. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → Fun 𝐴)

Theoremajval 28637* Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑄 = (·𝑖OLD𝑊)    &   𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))

18.6  Complex Banach spaces

18.6.1  Definition and basic properties

Syntaxccbn 28638 Extend class notation with the class of all complex Banach spaces.
class CBan

Definitiondf-cbn 28639 Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 23938 instead. (New usage is discouraged.)
CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))}

Theoremiscbn 28640 A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 23940 instead. (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋)))

Theoremcbncms 28641 The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 23949 (or preferably bncms 23946) instead. (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))

Theorembnnv 28642 Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 23942 instead. (New usage is discouraged.)
(𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)

Theorembnrel 28643 The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CBan

Theorembnsscmcl 28644 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 ∧ 𝑊𝐻) → (𝑊 ∈ CBan ↔ 𝑌 ∈ (Clsd‘𝐽)))

18.6.2  Examples of complex Banach spaces

Theoremcnbn 28645 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

Theoremubthlem1 28646* Lemma for ubth 28649. 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 23932, for some 𝑛 the set 𝐴𝑛 has an interior, meaning that there is a closed ball {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))    &   (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)    &   𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})       (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))

Theoremubthlem2 28647* Lemma for ubth 28649. Given that there is a closed ball 𝐵(𝑃, 𝑅) in 𝐴𝐾, for any 𝑥𝐵(0, 1), we have 𝑃 + 𝑅 · 𝑥𝐵(𝑃, 𝑅) and 𝑃𝐵(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 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))    &   (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)    &   𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑃𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝐴𝐾))       (𝜑 → ∃𝑑 ∈ ℝ ∀𝑡𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑)

Theoremubthlem3 28648* Lemma for ubth 28649. Prove the reverse implication, using nmblolbi 28576. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑊)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CBan    &   𝑊 ∈ NrmCVec    &   (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))       (𝜑 → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡𝑇 ((𝑈 normOpOLD 𝑊)‘𝑡) ≤ 𝑑))

Theoremubth 28649* 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 = (normCV𝑊)    &   𝑀 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ CBan ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ⊆ (𝑈 BLnOp 𝑊)) → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 ↔ ∃𝑑 ∈ ℝ ∀𝑡𝑇 (𝑀𝑡) ≤ 𝑑))

18.6.4  Minimizing Vector Theorem

Theoremminvecolem1 28650* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))       (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤𝑅 0 ≤ 𝑤))

Theoremminvecolem2 28651* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑𝐾𝑌)    &   (𝜑𝐿𝑌)    &   (𝜑 → ((𝐴𝐷𝐾)↑2) ≤ ((𝑆↑2) + 𝐵))    &   (𝜑 → ((𝐴𝐷𝐿)↑2) ≤ ((𝑆↑2) + 𝐵))       (𝜑 → ((𝐾𝐷𝐿)↑2) ≤ (4 · 𝐵))

Theoremminvecolem3 28652* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝐹 ∈ (Cau‘𝐷))

Theoremminvecolem4a 28653* Lemma for minveco 28660. 𝐹 is convergent in the subspace topology on 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))

Theoremminvecolem4b 28654* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑 → ((⇝𝑡𝐽)‘𝐹) ∈ 𝑋)

Theoremminvecolem4c 28655* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))       (𝜑𝑆 ∈ ℝ)

Theoremminvecolem4 28656* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )    &   (𝜑𝐹:ℕ⟶𝑌)    &   ((𝜑𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))    &   𝑇 = (1 / (((((𝐴𝐷((⇝𝑡𝐽)‘𝐹)) + 𝑆) / 2)↑2) − (𝑆↑2)))       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminvecolem5 28657* Lemma for minveco 28660. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9856. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑 → ∃𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminvecolem6 28658* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       ((𝜑𝑥𝑌) → (((𝐴𝐷𝑥)↑2) ≤ ((𝑆↑2) + 0) ↔ ∀𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))))

Theoremminvecolem7 28659* Lemma for minveco 28660. 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑅 = ran (𝑦𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))    &   𝑆 = inf(𝑅, ℝ, < )       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Theoremminveco 28660* 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 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   (𝜑𝑈 ∈ CPreHilOLD)    &   (𝜑𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))    &   (𝜑𝐴𝑋)       (𝜑 → ∃!𝑥𝑌𝑦𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

18.7  Complex Hilbert spaces

18.7.1  Definition and basic properties

Syntaxchlo 28661 Extend class notation with the class of all complex Hilbert spaces.
class CHilOLD

Definitiondf-hlo 28662 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)

Theoremishlo 28663 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))

Theoremhlobn 28664 Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ CBan)

Theoremhlph 28665 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 ∈ CHilOLD𝑈 ∈ CPreHilOLD)

Theoremhlrel 28666 The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CHilOLD

Theoremhlnv 28667 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
(𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)

Theoremhlnvi 28668 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
𝑈 ∈ CHilOLD       𝑈 ∈ NrmCVec

Theoremhlvc 28669 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑊 = (1st𝑈)       (𝑈 ∈ CHilOLD𝑊 ∈ CVecOLD)

Theoremhlcmet 28670 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 ∈ CHilOLD𝐷 ∈ (CMet‘𝑋))

Theoremhlmet 28671 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 ∈ CHilOLD𝐷 ∈ (Met‘𝑋))

Theoremhlpar2 28672 The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremhlpar 28673 The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

18.7.2  Standard axioms for a complex Hilbert space

Theoremhlex 28674 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

Theoremhladdf 28675 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       (𝑈 ∈ CHilOLD𝐺:(𝑋 × 𝑋)⟶𝑋)

Theoremhlcom 28676 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremhlass 28677 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theoremhl0cl 28678 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)       (𝑈 ∈ CHilOLD𝑍𝑋)

Theoremhladdid 28679 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremhlmulf 28680 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ CHilOLD𝑆:(ℂ × 𝑋)⟶𝑋)

Theoremhlmulid 28681 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremhlmulass 28682 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theoremhldi 28683 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theoremhldir 28684 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theoremhlmul0 28685 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋) → (0𝑆𝐴) = 𝑍)

Theoremhlipf 28686 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       (𝑈 ∈ CHilOLD𝑃:(𝑋 × 𝑋)⟶ℂ)

Theoremhlipcj 28687 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝑃𝐵) = (∗‘(𝐵𝑃𝐴)))

Theoremhlipdir 28688 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremhlipass 28689 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremhlipgt0 28690 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 = (0vec𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CHilOLD𝐴𝑋𝐴𝑍) → 0 < (𝐴𝑃𝐴))

Theoremhlcompl 28691 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 ∈ CHilOLD𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡𝐽))

18.7.3  Examples of complex Hilbert spaces

Theoremcnchl 28692 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

Theoremhtthlem 28693* Lemma for htth 28694. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = ⟨𝑤𝑇(𝑧)⟩ = ⟨𝑇(𝑤) ∣ 𝑧 for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 28631, so by the Uniform Boundedness theorem ubth 28649, 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)) ≤ 𝑦𝑇(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 = (·𝑖OLD𝑈)    &   𝐿 = (𝑈 LnOp 𝑈)    &   𝐵 = (𝑈 BLnOp 𝑈)    &   𝑁 = (normCV𝑈)    &   𝑈 ∈ CHilOLD    &   𝑊 = ⟨⟨ + , · ⟩, abs⟩    &   (𝜑𝑇𝐿)    &   (𝜑 → ∀𝑥𝑋𝑦𝑋 (𝑥𝑃(𝑇𝑦)) = ((𝑇𝑥)𝑃𝑦))    &   𝐹 = (𝑧𝑋 ↦ (𝑤𝑋 ↦ (𝑤𝑃(𝑇𝑧))))    &   𝐾 = (𝐹 “ {𝑧𝑋 ∣ (𝑁𝑧) ≤ 1})       (𝜑𝑇𝐵)

Theoremhtth 28694* 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 = (·𝑖OLD𝑈)    &   𝐿 = (𝑈 LnOp 𝑈)    &   𝐵 = (𝑈 BLnOp 𝑈)       ((𝑈 ∈ CHilOLD𝑇𝐿 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑃(𝑇𝑦)) = ((𝑇𝑥)𝑃𝑦)) → 𝑇𝐵)

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 20847; note that df-hil 20847 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

Syntaxchba 28695 Extend class notation with Hilbert vector space.
class

Syntaxcva 28696 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 10539.
class +

Syntaxcsm 28697 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 ·

Syntaxcsp 28698 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 4573.
class ·ih

Syntaxcno 28699 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 norm

Syntaxc0v 28700 Extend class notation with zero vector in Hilbert space.
class 0

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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 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