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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dipsubdir 30101 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ππ΅)ππΆ) = ((π΄ππΆ) β (π΅ππΆ))) | ||
Theorem | dipsubdi 30102 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅ππΆ)) = ((π΄ππ΅) β (π΄ππΆ))) | ||
Theorem | pythi 30103 | The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space π. 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.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD & β’ π΄ β π & β’ π΅ β π β β’ ((π΄ππ΅) = 0 β ((πβ(π΄πΊπ΅))β2) = (((πβπ΄)β2) + ((πβπ΅)β2))) | ||
Theorem | siilem1 30104 | Lemma for sii 30107. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD & β’ π΄ β π & β’ π΅ β π & β’ π = ( βπ£ βπ) & β’ π = ( Β·π OLD βπ) & β’ πΆ β β & β’ (πΆ Β· (π΄ππ΅)) β β & β’ 0 β€ (πΆ Β· (π΄ππ΅)) β β’ ((π΅ππ΄) = (πΆ Β· ((πβπ΅)β2)) β (ββ((π΄ππ΅) Β· (πΆ Β· ((πβπ΅)β2)))) β€ ((πβπ΄) Β· (πβπ΅))) | ||
Theorem | siilem2 30105 | Lemma for sii 30107. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD & β’ π΄ β π & β’ π΅ β π & β’ π = ( βπ£ βπ) & β’ π = ( Β·π OLD βπ) β β’ ((πΆ β β β§ (πΆ Β· (π΄ππ΅)) β β β§ 0 β€ (πΆ Β· (π΄ππ΅))) β ((π΅ππ΄) = (πΆ Β· ((πβπ΅)β2)) β (ββ((π΄ππ΅) Β· (πΆ Β· ((πβπ΅)β2)))) β€ ((πβπ΄) Β· (πβπ΅)))) | ||
Theorem | siii 30106 | Inference from sii 30107. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD & β’ π΄ β π & β’ π΅ β π β β’ (absβ(π΄ππ΅)) β€ ((πβπ΄) Β· (πβπ΅)) | ||
Theorem | sii 30107 | Obsolete version of ipcau 24755 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 30373, bcsiALT 30432, bcsiHIL 30433, csbren 24916. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD β β’ ((π΄ β π β§ π΅ β π) β (absβ(π΄ππ΅)) β€ ((πβπ΄) Β· (πβπ΅))) | ||
Theorem | ipblnfi 30108* | A function πΉ generated by varying the first argument of an inner product (with its second argument a fixed vector π΄) is a bounded linear functional, i.e. a bounded linear operator from the vector space to β. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD & β’ πΆ = β¨β¨ + , Β· β©, absβ© & β’ π΅ = (π BLnOp πΆ) & β’ πΉ = (π₯ β π β¦ (π₯ππ΄)) β β’ (π΄ β π β πΉ β π΅) | ||
Theorem | ip2eqi 30109* | Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD β β’ ((π΄ β π β§ π΅ β π) β (βπ₯ β π (π₯ππ΄) = (π₯ππ΅) β π΄ = π΅)) | ||
Theorem | phoeqi 30110* | A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD β β’ ((π:πβΆπ β§ π:πβΆπ) β (βπ₯ β π βπ¦ β π (π₯π(πβπ¦)) = (π₯π(πβπ¦)) β π = π)) | ||
Theorem | ajmoi 30111* | Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ π β CPreHilOLD β β’ β*π (π :πβΆπ β§ βπ₯ β π βπ¦ β π ((πβπ₯)ππ¦) = (π₯π(π βπ¦))) | ||
Theorem | ajfuni 30112 | The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
β’ π΄ = (πadjπ) & β’ π β CPreHilOLD & β’ π β NrmCVec β β’ Fun π΄ | ||
Theorem | ajfun 30113 | The adjoint function is a function. This is not immediately apparent from df-aj 30003 but results from the uniqueness shown by ajmoi 30111. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
β’ π΄ = (πadjπ) β β’ ((π β CPreHilOLD β§ π β NrmCVec) β Fun π΄) | ||
Theorem | ajval 30114* | Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ π = (Β·πOLDβπ) & β’ π΄ = (πadjπ) β β’ ((π β CPreHilOLD β§ π β NrmCVec β§ π:πβΆπ) β (π΄βπ) = (β©π (π :πβΆπ β§ βπ₯ β π βπ¦ β π ((πβπ₯)ππ¦) = (π₯π(π βπ¦))))) | ||
Syntax | ccbn 30115 | Extend class notation with the class of all complex Banach spaces. |
class CBan | ||
Definition | df-cbn 30116 | Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 24853 instead. (New usage is discouraged.) |
β’ CBan = {π’ β NrmCVec β£ (IndMetβπ’) β (CMetβ(BaseSetβπ’))} | ||
Theorem | iscbn 30117 | A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 24855 instead. (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β CBan β (π β NrmCVec β§ π· β (CMetβπ))) | ||
Theorem | cbncms 30118 | The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) Use bncmet 24864 (or preferably bncms 24861) instead. (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β CBan β π· β (CMetβπ)) | ||
Theorem | bnnv 30119 | Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) Use bnnvc 24857 instead. (New usage is discouraged.) |
β’ (π β CBan β π β NrmCVec) | ||
Theorem | bnrel 30120 | The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
β’ Rel CBan | ||
Theorem | bnsscmcl 30121 | 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.) |
β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π» = (SubSpβπ) & β’ π = (BaseSetβπ) β β’ ((π β CBan β§ π β π») β (π β CBan β π β (Clsdβπ½))) | ||
Theorem | cnbn 30122 | The set of complex numbers is a complex Banach space. (Contributed by Steve Rodriguez, 4-Jan-2007.) (New usage is discouraged.) |
β’ π = β¨β¨ + , Β· β©, absβ© β β’ π β CBan | ||
Theorem | ubthlem1 30123* | Lemma for ubth 30126. The function π΄ exhibits a countable collection of sets that are closed, being the inverse image under π‘ of the closed ball of radius π, and by assumption they cover π. Thus, by the Baire Category theorem bcth2 24847, for some π the set π΄βπ has an interior, meaning that there is a closed ball {π§ β π β£ (π¦π·π§) β€ π} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π β CBan & β’ π β NrmCVec & β’ (π β π β (π BLnOp π)) & β’ (π β βπ₯ β π βπ β β βπ‘ β π (πβ(π‘βπ₯)) β€ π) & β’ π΄ = (π β β β¦ {π§ β π β£ βπ‘ β π (πβ(π‘βπ§)) β€ π}) β β’ (π β βπ β β βπ¦ β π βπ β β+ {π§ β π β£ (π¦π·π§) β€ π} β (π΄βπ)) | ||
Theorem | ubthlem2 30124* | Lemma for ubth 30126. Given that there is a closed ball π΅(π, π ) in π΄βπΎ, for any π₯ β π΅(0, 1), we have π + π Β· π₯ β π΅(π, π ) and π β π΅(π, π ), so both of these have norm(π‘(π§)) β€ πΎ and so norm(π‘(π₯ )) β€ (norm(π‘(π)) + norm(π‘(π + π Β· π₯))) / π β€ ( πΎ + πΎ) / π , which is our desired uniform bound. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π β CBan & β’ π β NrmCVec & β’ (π β π β (π BLnOp π)) & β’ (π β βπ₯ β π βπ β β βπ‘ β π (πβ(π‘βπ₯)) β€ π) & β’ π΄ = (π β β β¦ {π§ β π β£ βπ‘ β π (πβ(π‘βπ§)) β€ π}) & β’ (π β πΎ β β) & β’ (π β π β π) & β’ (π β π β β+) & β’ (π β {π§ β π β£ (ππ·π§) β€ π } β (π΄βπΎ)) β β’ (π β βπ β β βπ‘ β π ((π normOpOLD π)βπ‘) β€ π) | ||
Theorem | ubthlem3 30125* | Lemma for ubth 30126. Prove the reverse implication, using nmblolbi 30053. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π β CBan & β’ π β NrmCVec & β’ (π β π β (π BLnOp π)) β β’ (π β (βπ₯ β π βπ β β βπ‘ β π (πβ(π‘βπ₯)) β€ π β βπ β β βπ‘ β π ((π normOpOLD π)βπ‘) β€ π)) | ||
Theorem | ubth 30126* | Uniform Boundedness Theorem, also called the Banach-Steinhaus Theorem. Let π be a collection of bounded linear operators on a Banach space. If, for every vector π₯, 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.) |
β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (π normOpOLD π) β β’ ((π β CBan β§ π β NrmCVec β§ π β (π BLnOp π)) β (βπ₯ β π βπ β β βπ‘ β π (πβ(π‘βπ₯)) β€ π β βπ β β βπ‘ β π (πβπ‘) β€ π)) | ||
Theorem | minvecolem1 30127* | Lemma for minveco 30137. The set of all distances from points of π to π΄ are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-May-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) β β’ (π β (π β β β§ π β β β§ βπ€ β π 0 β€ π€)) | ||
Theorem | minvecolem2 30128* | Lemma for minveco 30137. Any two points πΎ and πΏ in π are close to each other if they are close to the infimum of distance to π΄. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β π΅ β β) & β’ (π β 0 β€ π΅) & β’ (π β πΎ β π) & β’ (π β πΏ β π) & β’ (π β ((π΄π·πΎ)β2) β€ ((πβ2) + π΅)) & β’ (π β ((π΄π·πΏ)β2) β€ ((πβ2) + π΅)) β β’ (π β ((πΎπ·πΏ)β2) β€ (4 Β· π΅)) | ||
Theorem | minvecolem3 30129* | Lemma for minveco 30137. 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.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β πΉ:ββΆπ) & β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) β β’ (π β πΉ β (Cauβπ·)) | ||
Theorem | minvecolem4a 30130* | Lemma for minveco 30137. πΉ is convergent in the subspace topology on π. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β πΉ:ββΆπ) & β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) β β’ (π β πΉ(βπ‘β(MetOpenβ(π· βΎ (π Γ π))))((βπ‘β(MetOpenβ(π· βΎ (π Γ π))))βπΉ)) | ||
Theorem | minvecolem4b 30131* | Lemma for minveco 30137. The convergent point of the cauchy sequence πΉ is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β πΉ:ββΆπ) & β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) β β’ (π β ((βπ‘βπ½)βπΉ) β π) | ||
Theorem | minvecolem4c 30132* | Lemma for minveco 30137. The infimum of the distances to π΄ is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β πΉ:ββΆπ) & β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) β β’ (π β π β β) | ||
Theorem | minvecolem4 30133* | Lemma for minveco 30137. The convergent point of the cauchy sequence πΉ attains the minimum distance, and so is closer to π΄ than any other point in π. (Contributed by Mario Carneiro, 7-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) & β’ (π β πΉ:ββΆπ) & β’ ((π β§ π β β) β ((π΄π·(πΉβπ))β2) β€ ((πβ2) + (1 / π))) & β’ π = (1 / (((((π΄π·((βπ‘βπ½)βπΉ)) + π) / 2)β2) β (πβ2))) β β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) | ||
Theorem | minvecolem5 30134* | Lemma for minveco 30137. Discharge the assumption about the sequence πΉ by applying countable choice ax-cc 10430. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) β β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) | ||
Theorem | minvecolem6 30135* | Lemma for minveco 30137. Any minimal point is less than π away from π΄. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) β β’ ((π β§ π₯ β π) β (((π΄π·π₯)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)))) | ||
Theorem | minvecolem7 30136* | Lemma for minveco 30137. 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.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) & β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) & β’ π = ran (π¦ β π β¦ (πβ(π΄ππ¦))) & β’ π = inf(π , β, < ) β β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) | ||
Theorem | minveco 30137* | Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace π that minimizes the distance to an arbitrary vector π΄ 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.) |
β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π = (BaseSetβπ) & β’ (π β π β CPreHilOLD) & β’ (π β π β ((SubSpβπ) β© CBan)) & β’ (π β π΄ β π) β β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) | ||
Syntax | chlo 30138 | Extend class notation with the class of all complex Hilbert spaces. |
class CHilOLD | ||
Definition | df-hlo 30139 | 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.) |
β’ CHilOLD = (CBan β© CPreHilOLD) | ||
Theorem | ishlo 30140 | 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.) |
β’ (π β CHilOLD β (π β CBan β§ π β CPreHilOLD)) | ||
Theorem | hlobn 30141 | Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
β’ (π β CHilOLD β π β CBan) | ||
Theorem | hlph 30142 | 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.) |
β’ (π β CHilOLD β π β CPreHilOLD) | ||
Theorem | hlrel 30143 | The class of all complex Hilbert spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
β’ Rel CHilOLD | ||
Theorem | hlnv 30144 | Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
β’ (π β CHilOLD β π β NrmCVec) | ||
Theorem | hlnvi 30145 | Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
β’ π β CHilOLD β β’ π β NrmCVec | ||
Theorem | hlvc 30146 | Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (1st βπ) β β’ (π β CHilOLD β π β CVecOLD) | ||
Theorem | hlcmet 30147 | The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β CHilOLD β π· β (CMetβπ)) | ||
Theorem | hlmet 30148 | The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β CHilOLD β π· β (Metβπ)) | ||
Theorem | hlpar2 30149 | The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) β β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (((πβ(π΄πΊπ΅))β2) + ((πβ(π΄ππ΅))β2)) = (2 Β· (((πβπ΄)β2) + ((πβπ΅)β2)))) | ||
Theorem | hlpar 30150 | The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (((πβ(π΄πΊπ΅))β2) + ((πβ(π΄πΊ(-1ππ΅)))β2)) = (2 Β· (((πβπ΄)β2) + ((πβπ΅)β2)))) | ||
Theorem | hlex 30151 | The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) β β’ π β V | ||
Theorem | hladdf 30152 | Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) β β’ (π β CHilOLD β πΊ:(π Γ π)βΆπ) | ||
Theorem | hlcom 30153 | Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) β β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (π΄πΊπ΅) = (π΅πΊπ΄)) | ||
Theorem | hlass 30154 | Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) β β’ ((π β CHilOLD β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)πΊπΆ) = (π΄πΊ(π΅πΊπΆ))) | ||
Theorem | hl0cl 30155 | The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (0vecβπ) β β’ (π β CHilOLD β π β π) | ||
Theorem | hladdid 30156 | Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = (0vecβπ) β β’ ((π β CHilOLD β§ π΄ β π) β (π΄πΊπ) = π΄) | ||
Theorem | hlmulf 30157 | Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) β β’ (π β CHilOLD β π:(β Γ π)βΆπ) | ||
Theorem | hlmulid 30158 | Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β CHilOLD β§ π΄ β π) β (1ππ΄) = π΄) | ||
Theorem | hlmulass 30159 | Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β β β§ πΆ β π)) β ((π΄ Β· π΅)ππΆ) = (π΄π(π΅ππΆ))) | ||
Theorem | hldi 30160 | Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅πΊπΆ)) = ((π΄ππ΅)πΊ(π΄ππΆ))) | ||
Theorem | hldir 30161 | Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β β β§ πΆ β π)) β ((π΄ + π΅)ππΆ) = ((π΄ππΆ)πΊ(π΅ππΆ))) | ||
Theorem | hlmul0 30162 | Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) β β’ ((π β CHilOLD β§ π΄ β π) β (0ππ΄) = π) | ||
Theorem | hlipf 30163 | Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ (π β CHilOLD β π:(π Γ π)βΆβ) | ||
Theorem | hlipcj 30164 | Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (ββ(π΅ππ΄))) | ||
Theorem | hlipdir 30165 | Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CHilOLD β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)ππΆ) = ((π΄ππΆ) + (π΅ππΆ))) | ||
Theorem | hlipass 30166 | Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β ((π΄ππ΅)ππΆ) = (π΄ Β· (π΅ππΆ))) | ||
Theorem | hlipgt0 30167 | The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β CHilOLD β§ π΄ β π β§ π΄ β π) β 0 < (π΄ππ΄)) | ||
Theorem | hlcompl 30168 | Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
β’ π· = (IndMetβπ) & β’ π½ = (MetOpenβπ·) β β’ ((π β CHilOLD β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) | ||
Theorem | cnchl 30169 | The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
β’ π = β¨β¨ + , Β· β©, absβ© β β’ π β CHilOLD | ||
Theorem | htthlem 30170* | Lemma for htth 30171. The collection πΎ, which consists of functions πΉ(π§)(π€) = β¨π€ β£ π(π§)β© = β¨π(π€) β£ π§β© for each π§ in the unit ball, is a collection of bounded linear functions by ipblnfi 30108, so by the Uniform Boundedness theorem ubth 30126, there is a uniform bound π¦ on β₯ πΉ(π₯) β₯ for all π₯ in the unit ball. Then β£ π(π₯) β£ β2 = β¨π(π₯) β£ π(π₯)β© = πΉ(π₯)( π(π₯)) β€ π¦ β£ π(π₯) β£, so β£ π(π₯) β£ β€ π¦ and π is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ πΏ = (π LnOp π) & β’ π΅ = (π BLnOp π) & β’ π = (normCVβπ) & β’ π β CHilOLD & β’ π = β¨β¨ + , Β· β©, absβ© & β’ (π β π β πΏ) & β’ (π β βπ₯ β π βπ¦ β π (π₯π(πβπ¦)) = ((πβπ₯)ππ¦)) & β’ πΉ = (π§ β π β¦ (π€ β π β¦ (π€π(πβπ§)))) & β’ πΎ = (πΉ β {π§ β π β£ (πβπ§) β€ 1}) β β’ (π β π β π΅) | ||
Theorem | htth 30171* | 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.) |
β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) & β’ πΏ = (π LnOp π) & β’ π΅ = (π BLnOp π) β β’ ((π β CHilOLD β§ π β πΏ β§ βπ₯ β π βπ¦ β π (π₯π(πβπ¦)) = ((πβπ₯)ππ¦)) β π β π΅) | ||
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 21259; note that df-hil 21259 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. | ||
Syntax | chba 30172 | Extend class notation with Hilbert vector space. |
class β | ||
Syntax | cva 30173 | 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 11113. |
class +β | ||
Syntax | csm 30174 | 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 Β·β | ||
Syntax | csp 30175 | Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of π΄ and π΅ is usually written β¨π΄, π΅β© but our operation notation allows to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4636. |
class Β·ih | ||
Syntax | cno 30176 | Extend class notation with the norm function in Hilbert space. In the literature, the norm of π΄ is usually written "|| π΄ ||", but we use function notation to take advantage of our existing theorems about functions. |
class normβ | ||
Syntax | c0v 30177 | Extend class notation with zero vector in Hilbert space. |
class 0β | ||
Syntax | cmv 30178 | Extend class notation with vector subtraction in Hilbert space. |
class ββ | ||
Syntax | ccauold 30179 | Extend class notation with set of Cauchy sequences in Hilbert space. |
class Cauchy | ||
Syntax | chli 30180 | Extend class notation with convergence relation in Hilbert space. |
class βπ£ | ||
Syntax | csh 30181 | Extend class notation with set of subspaces of a Hilbert space. |
class Sβ | ||
Syntax | cch 30182 | Extend class notation with set of closed subspaces of a Hilbert space. |
class Cβ | ||
Syntax | cort 30183 | Extend class notation with orthogonal complement in Cβ. |
class β₯ | ||
Syntax | cph 30184 | Extend class notation with subspace sum in Cβ. |
class +β | ||
Syntax | cspn 30185 | Extend class notation with subspace span in Cβ. |
class span | ||
Syntax | chj 30186 | Extend class notation with join in Cβ. |
class β¨β | ||
Syntax | chsup 30187 | Extend class notation with supremum of a collection in Cβ. |
class β¨β | ||
Syntax | c0h 30188 | Extend class notation with zero of Cβ. |
class 0β | ||
Syntax | ccm 30189 | Extend class notation with the commutes relation on a Hilbert lattice. |
class πΆβ | ||
Syntax | cpjh 30190 | Extend class notation with set of projections on a Hilbert space. |
class projβ | ||
Syntax | chos 30191 | Extend class notation with sum of Hilbert space operators. |
class +op | ||
Syntax | chot 30192 | Extend class notation with scalar product of a Hilbert space operator. |
class Β·op | ||
Syntax | chod 30193 | Extend class notation with difference of Hilbert space operators. |
class βop | ||
Syntax | chfs 30194 | Extend class notation with sum of Hilbert space functionals. |
class +fn | ||
Syntax | chft 30195 | Extend class notation with scalar product of Hilbert space functional. |
class Β·fn | ||
Syntax | ch0o 30196 | Extend class notation with the Hilbert space zero operator. |
class 0hop | ||
Syntax | chio 30197 | Extend class notation with Hilbert space identity operator. |
class Iop | ||
Syntax | cnop 30198 | Extend class notation with the operator norm function. |
class normop | ||
Syntax | ccop 30199 | Extend class notation with set of continuous Hilbert space operators. |
class ContOp | ||
Syntax | clo 30200 | Extend class notation with set of linear Hilbert space operators. |
class LinOp |
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