 Home Metamath Proof ExplorerTheorem List (p. 201 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremremulg 20001 The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ) → (𝑁(.g‘ℝfld)𝐴) = (𝑁 · 𝐴))

Theoremresubdrg 20002 The real numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)
(ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing)

Theoremresubgval 20003 Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
= (-g‘ℝfld)       ((𝑋 ∈ ℝ ∧ 𝑌 ∈ ℝ) → (𝑋𝑌) = (𝑋 𝑌))

Theoremreplusg 20004 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
+ = (+g‘ℝfld)

Theoremremulr 20005 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
· = (.r‘ℝfld)

Theoremre0g 20006 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
0 = (0g‘ℝfld)

Theoremre1r 20007 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
1 = (1r‘ℝfld)

Theoremrele2 20008 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
≤ = (le‘ℝfld)

Theoremrelt 20009 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
< = (lt‘ℝfld)

Theoremreds 20010 The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
(abs ∘ − ) = (dist‘ℝfld)

Theoremredvr 20011 The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴(/r‘ℝfld)𝐵) = (𝐴 / 𝐵))

Theoremretos 20012 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ Toset

Theoremrefld 20013 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
fld ∈ Field

Theoremrefldcj 20014 The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
∗ = (*𝑟‘ℝfld)

Theoremrecrng 20015 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.)
fld ∈ *-Ring

Theoremregsumsupp 20016* The group sum over the real numbers, expressed as a finite sum. (Contributed by Thierry Arnoux, 22-Jun-2019.) (Proof shortened by AV, 19-Jul-2019.)
((𝐹:𝐼⟶ℝ ∧ 𝐹 finSupp 0 ∧ 𝐼𝑉) → (ℝfld Σg 𝐹) = Σ𝑥 ∈ (𝐹 supp 0)(𝐹𝑥))

10.12  Generalized pre-Hilbert and Hilbert spaces

10.12.1  Definition and basic properties

Syntaxcphl 20017 Extend class notation with class of all pre-Hilbert spaces.
class PreHil

Syntaxcipf 20018 Extend class notation with inner product function.
class ·if

Definitiondf-phl 20019* Define the class of all pre-Hilbert spaces (inner product spaces) over arbitrary fields with involution. (Some textbook definitions are more restrictive and require the field of scalars to be the field of real or complex numbers). (Contributed by NM, 22-Sep-2011.)
PreHil = {𝑔 ∈ LVec ∣ [(Base‘𝑔) / 𝑣][(·𝑖𝑔) / ][(Scalar‘𝑔) / 𝑓](𝑓 ∈ *-Ring ∧ ∀𝑥𝑣 ((𝑦𝑣 ↦ (𝑦𝑥)) ∈ (𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥𝑥) = (0g𝑓) → 𝑥 = (0g𝑔)) ∧ ∀𝑦𝑣 ((*𝑟𝑓)‘(𝑥𝑦)) = (𝑦𝑥)))}

Definitiondf-ipf 20020* Define the inner product function. Usually we will use ·𝑖 directly instead of ·if, and they have the same behavior in most cases. The main advantage of ·if is that it is a guaranteed function (ipffn 20044), while ·𝑖 only has closure (ipcl 20026). (Contributed by Mario Carneiro, 12-Aug-2015.)
·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))

Theoremisphl 20021* The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &    0 = (0g𝑊)    &    = (*𝑟𝐹)    &   𝑍 = (0g𝐹)       (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = 𝑍𝑥 = 0 ) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))

Theoremphllvec 20022 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Theoremphllmod 20023 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ PreHil → 𝑊 ∈ LMod)

Theoremphlsrng 20024 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Theoremphllmhm 20025* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Theoremipcl 20026 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ 𝐾)

Theoremipcj 20027 Conjugate of an inner product in a pre-Hilbert space. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremiporthcom 20028 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍))

Theoremip0l 20029 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 𝑍)

Theoremip0r 20030 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 𝑍)

Theoremipeq0 20031 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 𝑍𝐴 = 0 ))

Theoremipdir 20032 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) (𝐵 , 𝐶)))

Theoremipdi 20033 Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) (𝐴 , 𝐶)))

Theoremip2di 20034 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) (𝐵 , 𝐷)) ((𝐴 , 𝐷) (𝐵 , 𝐶))))

Theoremipsubdir 20035 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶)𝑆(𝐵 , 𝐶)))

Theoremipsubdi 20036 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)))

Theoremip2subdi 20037 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)    &    + = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 𝐵) , (𝐶 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))

Theoremipass 20038 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶)))

Theoremipassr 20039 "Associative" law for second argument of inner product (compare ipass 20038). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( 𝐶)))

Theoremipassr2 20040 "Associative" law for inner product. Conjugate version of ipassr 20039. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( 𝐶) · 𝐵)))

Theoremipffval 20041* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)        · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))

Theoremipfval 20042 The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Theoremipfeq 20043 If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ( , Fn (𝑉 × 𝑉) → · = , )

Theoremipffn 20044 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)        , Fn (𝑉 × 𝑉)

Theoremphlipf 20045 The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾)

Theoremip2eq 20046* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵)))

Theoremisphld 20047* Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐼 = (·𝑖𝑊))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐾 = (Base‘𝐹))    &   (𝜑 = (+g𝐹))    &   (𝜑× = (.r𝐹))    &   (𝜑 = (*𝑟𝐹))    &   (𝜑𝑂 = (0g𝐹))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐹 ∈ *-Ring)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)    &   ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))    &   ((𝜑𝑥𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )    &   ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))       (𝜑𝑊 ∈ PreHil)

Theoremphlpropd 20048* If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))       (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Theoremssipeq 20049 The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)       (𝑈𝑆𝑃 = , )

Theoremphssipval 20050 The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ (𝐴𝑈𝐵𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵))

Theoremphssip 20051 The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &    · = (·if𝑊)    &   𝑃 = (·if𝑋)       ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))

10.12.2  Orthocomplements and closed subspaces

Syntaxcocv 20052 Extend class notation with orthocomplement of a subspace.
class ocv

Syntaxccss 20053 Extend class notation with set of closed subspaces.
class CSubSp

Syntaxcthl 20054 Extend class notation with the Hilbert lattice.
class toHL

Definitiondf-ocv 20055* Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.)
ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))

Definitiondf-css 20056* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
CSubSp = ( ∈ V ↦ {𝑠𝑠 = ((ocv‘)‘((ocv‘)‘𝑠))})

Definitiondf-thl 20057 Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))

Theoremocvfval 20058* The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))

Theoremocvval 20059* Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })

Theoremelocv 20060* Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))

Theoremocvi 20061 Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )

Theoremocvss 20062 The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ( 𝑆) ⊆ 𝑉

Theoremocvocv 20063 A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → 𝑆 ⊆ ( ‘( 𝑆)))

Theoremocvlss 20064 The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐿)

Theoremocv2ss 20065 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)       (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))

Theoremocvin 20066 An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝐿) → (𝑆 ∩ ( 𝑆)) = { 0 })

Theoremocvsscon 20067 Two ways to say that 𝑆 and 𝑇 are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉𝑇𝑉) → (𝑆 ⊆ ( 𝑇) ↔ 𝑇 ⊆ ( 𝑆)))

Theoremocvlsp 20068 The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( ‘(𝑁𝑆)) = ( 𝑆))

Theoremocv0 20069 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ( ‘∅) = 𝑉

Theoremocvz 20070 The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → ( ‘{ 0 }) = 𝑉)

Theoremocv1 20071 The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → ( 𝑉) = { 0 })

Theoremunocv 20072 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)       ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Theoremiunocv 20073* The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       ( 𝑥𝐴 𝐵) = (𝑉 𝑥𝐴 ( 𝐵))

Theoremcssval 20074* The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})

Theoremiscss 20075 The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))

Theoremcssi 20076 Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑆𝐶𝑆 = ( ‘( 𝑆)))

Theoremcssss 20077 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑆𝐶𝑆𝑉)

Theoremiscss2 20078 It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝑆𝐶 ↔ ( ‘( 𝑆)) ⊆ 𝑆))

Theoremocvcss 20079 The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐶)

Theoremcssincl 20080 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremcss0 20081 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)

Theoremcss1 20082 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝑉𝐶)

Theoremcsslss 20083 A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝐶) → 𝑆𝐿)

Theoremlsmcss 20084 A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &   𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝑆𝑉)    &   (𝜑 → ( ‘( 𝑆)) ⊆ (𝑆 ( 𝑆)))       (𝜑𝑆𝐶)

Theoremcssmre 20085 The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 16296: consider the Hilbert space of sequences ℕ⟶ℝ with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 16362. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))

Theoremmrccss 20086 The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &   𝐹 = (mrCls‘𝐶)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝐹𝑆) = ( ‘( 𝑆)))

Theoremthlval 20087 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (ocv‘𝑊)       (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))

Theoremthlbas 20088 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       𝐶 = (Base‘𝐾)

Theoremthlle 20089 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (le‘𝐼)        = (le‘𝐾)

Theoremthlleval 20090 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &    = (le‘𝐾)       ((𝑆𝐶𝑇𝐶) → (𝑆 𝑇𝑆𝑇))

Theoremthloc 20091 Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &    = (ocv‘𝑊)        = (oc‘𝐾)

10.12.3  Orthogonal projection and orthonormal bases

Syntaxcpj 20092 Extend class notation with orthogonal projection function.
class proj

Syntaxchs 20093 Extend class notation with class of all Hilbert spaces.
class Hil

Syntaxcobs 20094 Extend class notation with the set of orthonormal bases.
class OBasis

Definitiondf-pj 20095* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18098, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
proj = ( ∈ V ↦ ((𝑥 ∈ (LSubSp‘) ↦ (𝑥(proj1)((ocv‘)‘𝑥))) ∩ (V × ((Base‘) ↑𝑚 (Base‘)))))

Definitiondf-hil 20096 Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.)
Hil = { ∈ PreHil ∣ dom (proj‘) = (CSubSp‘)}

Definitiondf-obs 20097* Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})

Theorempjfval 20098* The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))

Theorempjdm 20099 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))

Theorempjpm 20100 The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 ∈ ((𝑉𝑚 𝑉) ↑pm 𝐿)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >