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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremncvsprp 24901 Proportionality property of the norm of a scalar product in a normed subcomplex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ (NrmVec ∩ β„‚Vec) ∧ 𝐴 ∈ 𝐾 ∧ 𝐡 ∈ 𝑉) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = ((absβ€˜π΄) Β· (π‘β€˜π΅)))
 
Theoremncvsge0 24902 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ (NrmVec ∩ β„‚Vec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≀ 𝐴) ∧ 𝐡 ∈ 𝑉) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = (𝐴 Β· (π‘β€˜π΅)))
 
Theoremncvsm1 24903 The norm of the opposite of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’   ((π‘Š ∈ (NrmVec ∩ β„‚Vec) ∧ 𝐴 ∈ 𝑉) β†’ (π‘β€˜(-1 Β· 𝐴)) = (π‘β€˜π΄))
 
Theoremncvsdif 24904 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ (NrmVec ∩ β„‚Vec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (π‘β€˜(𝐴 + (-1 Β· 𝐡))) = (π‘β€˜(𝐡 + (-1 Β· 𝐴))))
 
Theoremncvspi 24905 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ (NrmVec ∩ β„‚Vec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ i ∈ 𝐾) β†’ (π‘β€˜(𝐴 + (i Β· 𝐡))) = (π‘β€˜(𝐡 + (-i Β· 𝐴))))
 
Theoremncvs1 24906 From any nonzero vector of a normed subcomplex vector space, construct a collinear vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    Β· = ( ·𝑠 β€˜πΊ)    &   πΉ = (Scalarβ€˜πΊ)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((𝐺 ∈ (NrmVec ∩ β„‚Vec) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 β‰  0 ) ∧ (1 / (π‘β€˜π΄)) ∈ 𝐾) β†’ (π‘β€˜((1 / (π‘β€˜π΄)) Β· 𝐴)) = 1)
 
Theoremcnrnvc 24907 The module of complex numbers (as a module over itself) is a normed vector space over itself. The vector operation is +, and the scalar product is Β·, and the norm function is abs. (Contributed by AV, 9-Oct-2021.)
𝐢 = (ringLModβ€˜β„‚fld)    β‡’   πΆ ∈ NrmVec
 
Theoremcnncvs 24908 The module of complex numbers (as a module over itself) is a normed subcomplex vector space. The vector operation is +, the scalar product is Β·, and the norm is abs (see cnnm 24909) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
𝐢 = (ringLModβ€˜β„‚fld)    β‡’   πΆ ∈ (NrmVec ∩ β„‚Vec)
 
Theoremcnnm 24909 The norm of the normed subcomplex vector space of complex numbers is the absolute value. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.)
𝐢 = (ringLModβ€˜β„‚fld)    β‡’   (normβ€˜πΆ) = abs
 
Theoremncvspds 24910 Value of the distance function in terms of the norm of a normed subcomplex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.)
𝑁 = (normβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &    Β· = ( ·𝑠 β€˜πΊ)    β‡’   ((𝐺 ∈ (NrmVec ∩ β„‚Vec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (π‘β€˜(𝐴 + (-1 Β· 𝐡))))
 
Theoremcnindmet 24911 The metric induced on the complex numbers. cnmet 24509 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 21155 and also cnmet 24509. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
𝑇 = (β„‚fld toNrmGrp abs)    β‡’   (distβ€˜π‘‡) = (abs ∘ βˆ’ )
 
Theoremcnncvsaddassdemo 24912 Derive the associative law for complex number addition addass 11201 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 + 𝐡) + 𝐢) = (𝐴 + (𝐡 + 𝐢)))
 
Theoremcnncvsmulassdemo 24913 Derive the associative law for complex number multiplication mulass 11202 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐴 Β· 𝐡) Β· 𝐢) = (𝐴 Β· (𝐡 Β· 𝐢)))
 
Theoremcnncvsabsnegdemo 24914 Derive the absolute value of a negative complex number absneg 15229 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
(𝐴 ∈ β„‚ β†’ (absβ€˜-𝐴) = (absβ€˜π΄))
 
12.5.4  Subcomplex pre-Hilbert spaces
 
Syntaxccph 24915 Extend class notation with the class of subcomplex pre-Hilbert spaces.
class β„‚PreHil
 
Syntaxctcph 24916 Function to put a norm on a pre-Hilbert space.
class toβ„‚PreHil
 
Definitiondf-cph 24917* Define the class of subcomplex pre-Hilbert spaces. By restricting the scalar field to a subfield of β„‚fld closed under square roots of nonnegative reals, we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
β„‚PreHil = {𝑀 ∈ (PreHil ∩ NrmMod) ∣ [(Scalarβ€˜π‘€) / 𝑓][(Baseβ€˜π‘“) / π‘˜](𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))))}
 
Definitiondf-tcph 24918* Define a function to augment a pre-Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed subcomplex pre-Hilbert space (see tcphcph 24986). (Contributed by Mario Carneiro, 7-Oct-2015.)
toβ„‚PreHil = (𝑀 ∈ V ↦ (𝑀 toNrmGrp (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))))
 
Theoremiscph 24919* A subcomplex pre-Hilbert space is exactly a pre-Hilbert space over a subfield of the field of complex numbers closed under square roots of nonnegative reals equipped with a norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
 
Theoremcphphl 24920 A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ PreHil)
 
Theoremcphnlm 24921 A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmMod)
 
Theoremcphngp 24922 A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmGrp)
 
Theoremcphlmod 24923 A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ LMod)
 
Theoremcphlvec 24924 A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ LVec)
 
Theoremcphnvc 24925 A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmVec)
 
Theoremcphsubrglem 24926 Lemma for cphsubrg 24929. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝐾 = (Baseβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐹 ∈ DivRing)    β‡’   (πœ‘ β†’ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ 𝐾 = (𝐴 ∩ β„‚) ∧ 𝐾 ∈ (SubRingβ€˜β„‚fld)))
 
Theoremcphreccllem 24927 Lemma for cphreccl 24930. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Baseβ€˜πΉ)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐴))    &   (πœ‘ β†’ 𝐹 ∈ DivRing)    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 β‰  0) β†’ (1 / 𝑋) ∈ 𝐾)
 
Theoremcphsca 24928 A subcomplex pre-Hilbert space is a vector space over a subfield of β„‚fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚PreHil β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))
 
Theoremcphsubrg 24929 The scalar field of a subcomplex pre-Hilbert space is a subring of β„‚fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚PreHil β†’ 𝐾 ∈ (SubRingβ€˜β„‚fld))
 
Theoremcphreccl 24930 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 β‰  0) β†’ (1 / 𝐴) ∈ 𝐾)
 
Theoremcphdivcl 24931 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐡 ∈ 𝐾 ∧ 𝐡 β‰  0)) β†’ (𝐴 / 𝐡) ∈ 𝐾)
 
Theoremcphcjcl 24932 The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝐾) β†’ (βˆ—β€˜π΄) ∈ 𝐾)
 
Theoremcphsqrtcl 24933 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of nonnegative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≀ 𝐴)) β†’ (βˆšβ€˜π΄) ∈ 𝐾)
 
Theoremcphabscl 24934 The scalar field of a subcomplex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝐾) β†’ (absβ€˜π΄) ∈ 𝐾)
 
Theoremcphsqrtcl2 24935 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝐾 ∧ Β¬ -𝐴 ∈ ℝ+) β†’ (βˆšβ€˜π΄) ∈ 𝐾)
 
Theoremcphsqrtcl3 24936 If the scalar field of a subcomplex pre-Hilbert space contains the imaginary unit i, then it is closed under square roots (i.e., it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) β†’ (βˆšβ€˜π΄) ∈ 𝐾)
 
Theoremcphqss 24937 The scalar field of a subcomplex pre-Hilbert space contains the rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚PreHil β†’ β„š βŠ† 𝐾)
 
Theoremcphclm 24938 A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ β„‚Mod)
 
Theoremcphnmvs 24939 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜(𝑋 Β· π‘Œ)) = ((absβ€˜π‘‹) Β· (π‘β€˜π‘Œ)))
 
Theoremcphipcl 24940 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴 , 𝐡) ∈ β„‚)
 
Theoremcphnmfval 24941* The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   (π‘Š ∈ β„‚PreHil β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
 
Theoremcphnm 24942 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ (π‘β€˜π΄) = (βˆšβ€˜(𝐴 , 𝐴)))
 
Theoremnmsq 24943 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ ((π‘β€˜π΄)↑2) = (𝐴 , 𝐴))
 
Theoremcphnmf 24944 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ β„‚PreHil β†’ 𝑁:π‘‰βŸΆπΎ)
 
Theoremcphnmcl 24945 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ (π‘β€˜π΄) ∈ 𝐾)
 
Theoremreipcl 24946 An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 , 𝐴) ∈ ℝ)
 
Theoremipge0 24947 The inner product in a subcomplex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ 0 ≀ (𝐴 , 𝐴))
 
Theoremcphipcj 24948 Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj 21407. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (βˆ—β€˜(𝐴 , 𝐡)) = (𝐡 , 𝐴))
 
Theoremcphipipcj 24949 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (Revised by AV, 19-Oct-2021.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ ((𝐴 , 𝐡) Β· (𝐡 , 𝐴)) = ((absβ€˜(𝐴 , 𝐡))↑2))
 
Theoremcphorthcom 24950 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 21408. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ ((𝐴 , 𝐡) = 0 ↔ (𝐡 , 𝐴) = 0))
 
Theoremcphip0l 24951 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 21409. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ ( 0 , 𝐴) = 0)
 
Theoremcphip0r 24952 Inner product with a zero second argument. Complex version of ip0r 21410. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ (𝐴 , 0 ) = 0)
 
Theoremcphipeq0 24953 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. Complex version of ipeq0 21411. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉) β†’ ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 ))
 
Theoremcphdir 24954 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 21412. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝐴 + 𝐡) , 𝐢) = ((𝐴 , 𝐢) + (𝐡 , 𝐢)))
 
Theoremcphdi 24955 Distributive law for inner product (left-distributivity). Complex version of ipdi 21413. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (𝐴 , (𝐡 + 𝐢)) = ((𝐴 , 𝐡) + (𝐴 , 𝐢)))
 
Theoremcph2di 24956 Distributive law for inner product. Complex version of ip2di 21414. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐡) , (𝐢 + 𝐷)) = (((𝐴 , 𝐢) + (𝐡 , 𝐷)) + ((𝐴 , 𝐷) + (𝐡 , 𝐢))))
 
Theoremcphsubdir 24957 Distributive law for inner product subtraction. Complex version of ipsubdir 21415. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝐴 βˆ’ 𝐡) , 𝐢) = ((𝐴 , 𝐢) βˆ’ (𝐡 , 𝐢)))
 
Theoremcphsubdi 24958 Distributive law for inner product subtraction. Complex version of ipsubdi 21416. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (𝐴 , (𝐡 βˆ’ 𝐢)) = ((𝐴 , 𝐡) βˆ’ (𝐴 , 𝐢)))
 
Theoremcph2subdi 24959 Distributive law for inner product subtraction. Complex version of ip2subdi 21417. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ 𝑉)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐴 βˆ’ 𝐡) , (𝐢 βˆ’ 𝐷)) = (((𝐴 , 𝐢) + (𝐡 , 𝐷)) βˆ’ ((𝐴 , 𝐷) + (𝐡 , 𝐢))))
 
Theoremcphass 24960 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 21418, his5 30607. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ ((𝐴 Β· 𝐡) , 𝐢) = (𝐴 Β· (𝐡 , 𝐢)))
 
Theoremcphassr 24961 "Associative" law for second argument of inner product (compare cphass 24960). See ipassr 21419, his52 . (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉)) β†’ (𝐡 , (𝐴 Β· 𝐢)) = ((βˆ—β€˜π΄) Β· (𝐡 , 𝐢)))
 
Theoremcph2ass 24962 Move scalar multiplication to outside of inner product. See his35 30609. (Contributed by Mario Carneiro, 17-Oct-2015.)
, = (Β·π‘–β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐡 ∈ 𝐾) ∧ (𝐢 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) β†’ ((𝐴 Β· 𝐢) , (𝐡 Β· 𝐷)) = ((𝐴 Β· (βˆ—β€˜π΅)) Β· (𝐢 , 𝐷)))
 
Theoremcphassi 24963 Associative law for the first argument of an inner product with scalar 𝑖. (Contributed by AV, 17-Oct-2021.)
𝑋 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((i Β· 𝐡) , 𝐴) = (i Β· (𝐡 , 𝐴)))
 
Theoremcphassir 24964 "Associative" law for the second argument of an inner product with scalar 𝑖. (Contributed by AV, 17-Oct-2021.)
𝑋 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 , (i Β· 𝐡)) = (-i Β· (𝐴 , 𝐡)))
 
Theoremcphpyth 24965 The pythagorean theorem for a subcomplex pre-Hilbert space. 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.) (Revised by SN, 22-Sep-2024.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    β‡’   ((πœ‘ ∧ (𝐴 , 𝐡) = 0) β†’ ((π‘β€˜(𝐴 + 𝐡))↑2) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))
 
Theoremtcphex 24966* Lemma for tcphbas 24968 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    β‡’   (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))) ∈ V
 
Theoremtcphval 24967* Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   πΊ = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
 
Theoremtcphbas 24968 The base set of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    β‡’   π‘‰ = (Baseβ€˜πΊ)
 
Theoremtchplusg 24969 The addition operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’    + = (+gβ€˜πΊ)
 
Theoremtcphsub 24970 The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’    βˆ’ = (-gβ€˜πΊ)
 
Theoremtcphmulr 24971 The ring operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &    Β· = (.rβ€˜π‘Š)    β‡’    Β· = (.rβ€˜πΊ)
 
Theoremtcphsca 24972 The scalar field of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    β‡’   πΉ = (Scalarβ€˜πΊ)
 
Theoremtcphvsca 24973 The scalar multiplication of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    β‡’    Β· = ( ·𝑠 β€˜πΊ)
 
Theoremtcphip 24974 The inner product of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &    Β· = (Β·π‘–β€˜π‘Š)    β‡’    Β· = (Β·π‘–β€˜πΊ)
 
Theoremtcphtopn 24975 The topology of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π· = (distβ€˜πΊ)    &   π½ = (TopOpenβ€˜πΊ)    β‡’   (π‘Š ∈ 𝑉 β†’ 𝐽 = (MetOpenβ€˜π·))
 
Theoremtcphphl 24976 Augmentation of a subcomplex pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space (because all the original components are the same). (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    β‡’   (π‘Š ∈ PreHil ↔ 𝐺 ∈ PreHil)
 
Theoremtchnmfval 24977* The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘ = (normβ€˜πΊ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   (π‘Š ∈ Grp β†’ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
 
Theoremtcphnmval 24978 The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘ = (normβ€˜πΊ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    β‡’   ((π‘Š ∈ Grp ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜π‘‹) = (βˆšβ€˜(𝑋 , 𝑋)))
 
Theoremcphtcphnm 24979 The norm of a norm-augmented subcomplex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   (π‘Š ∈ β„‚PreHil β†’ 𝑁 = (normβ€˜πΊ))
 
Theoremtcphds 24980 The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜π‘Š)    β‡’   (π‘Š ∈ Grp β†’ (𝑁 ∘ βˆ’ ) = (distβ€˜πΊ))
 
Theoremphclm 24981 A pre-Hilbert space whose field of scalars is a restriction of the field of complex numbers is a subcomplex module. TODO: redundant hypotheses. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    β‡’   (πœ‘ β†’ π‘Š ∈ β„‚Mod)
 
Theoremtcphcphlem3 24982 Lemma for tcphcph 24986: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    &    , = (Β·π‘–β€˜π‘Š)    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝑉) β†’ (𝑋 , 𝑋) ∈ ℝ)
 
Theoremipcau2 24983* The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space built from a pre-Hilbert space with certain properties. The main theorem is ipcau 24987. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    &    , = (Β·π‘–β€˜π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ π‘₯ ∈ ℝ ∧ 0 ≀ π‘₯)) β†’ (βˆšβ€˜π‘₯) ∈ 𝐾)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑉) β†’ 0 ≀ (π‘₯ , π‘₯))    &   πΎ = (Baseβ€˜πΉ)    &   π‘ = (normβ€˜πΊ)    &   πΆ = ((π‘Œ , 𝑋) / (π‘Œ , π‘Œ))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (absβ€˜(𝑋 , π‘Œ)) ≀ ((π‘β€˜π‘‹) Β· (π‘β€˜π‘Œ)))
 
Theoremtcphcphlem1 24984* Lemma for tcphcph 24986: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    &    , = (Β·π‘–β€˜π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ π‘₯ ∈ ℝ ∧ 0 ≀ π‘₯)) β†’ (βˆšβ€˜π‘₯) ∈ 𝐾)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑉) β†’ 0 ≀ (π‘₯ , π‘₯))    &   πΎ = (Baseβ€˜πΉ)    &    βˆ’ = (-gβ€˜π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (βˆšβ€˜((𝑋 βˆ’ π‘Œ) , (𝑋 βˆ’ π‘Œ))) ≀ ((βˆšβ€˜(𝑋 , 𝑋)) + (βˆšβ€˜(π‘Œ , π‘Œ))))
 
Theoremtcphcphlem2 24985* Lemma for tcphcph 24986: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    &    , = (Β·π‘–β€˜π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ π‘₯ ∈ ℝ ∧ 0 ≀ π‘₯)) β†’ (βˆšβ€˜π‘₯) ∈ 𝐾)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑉) β†’ 0 ≀ (π‘₯ , π‘₯))    &   πΎ = (Baseβ€˜πΉ)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝐾)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (βˆšβ€˜((𝑋 Β· π‘Œ) , (𝑋 Β· π‘Œ))) = ((absβ€˜π‘‹) Β· (βˆšβ€˜(π‘Œ , π‘Œ))))
 
Theoremtcphcph 24986* The standard definition of a norm turns any pre-Hilbert space over a subfield of β„‚fld closed under square roots of nonnegative reals into a subcomplex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toβ„‚PreHilβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ PreHil)    &   (πœ‘ β†’ 𝐹 = (β„‚fld β†Ύs 𝐾))    &    , = (Β·π‘–β€˜π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐾 ∧ π‘₯ ∈ ℝ ∧ 0 ≀ π‘₯)) β†’ (βˆšβ€˜π‘₯) ∈ 𝐾)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑉) β†’ 0 ≀ (π‘₯ , π‘₯))    β‡’   (πœ‘ β†’ 𝐺 ∈ β„‚PreHil)
 
Theoremipcau 24987 The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space. Part of Lemma 3.2-1(a) of [Kreyszig] p. 137. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 11-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (absβ€˜(𝑋 , π‘Œ)) ≀ ((π‘β€˜π‘‹) Β· (π‘β€˜π‘Œ)))
 
Theoremnmparlem 24988 Lemma for nmpar 24989. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ (((π‘β€˜(𝐴 + 𝐡))↑2) + ((π‘β€˜(𝐴 βˆ’ 𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
 
Theoremnmpar 24989 A subcomplex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (((π‘β€˜(𝐴 + 𝐡))↑2) + ((π‘β€˜(𝐴 βˆ’ 𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
 
Theoremcphipval2 24990 Value of the inner product expressed by the norm defined by it. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 , 𝐡) = (((((π‘β€˜(𝐴 + 𝐡))↑2) βˆ’ ((π‘β€˜(𝐴 βˆ’ 𝐡))↑2)) + (i Β· (((π‘β€˜(𝐴 + (i Β· 𝐡)))↑2) βˆ’ ((π‘β€˜(𝐴 βˆ’ (i Β· 𝐡)))↑2)))) / 4))
 
Theorem4cphipval2 24991 Four times the inner product value cphipval2 24990. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (4 Β· (𝐴 , 𝐡)) = ((((π‘β€˜(𝐴 + 𝐡))↑2) βˆ’ ((π‘β€˜(𝐴 βˆ’ 𝐡))↑2)) + (i Β· (((π‘β€˜(𝐴 + (i Β· 𝐡)))↑2) βˆ’ ((π‘β€˜(𝐴 βˆ’ (i Β· 𝐡)))↑2)))))
 
Theoremcphipval 24992* Value of the inner product expressed by a sum of terms with the norm defined by the inner product. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   (((π‘Š ∈ β„‚PreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴 , 𝐡) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴 + ((iβ†‘π‘˜) Β· 𝐡)))↑2)) / 4))
 
Theoremipcnlem2 24993 The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   π‘‡ = ((𝑅 / 2) / ((π‘β€˜π΄) + 1))    &   π‘ˆ = ((𝑅 / 2) / ((π‘β€˜π΅) + 𝑇))    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ ℝ+)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (𝐴𝐷𝑋) < π‘ˆ)    &   (πœ‘ β†’ (π΅π·π‘Œ) < 𝑇)    β‡’   (πœ‘ β†’ (absβ€˜((𝐴 , 𝐡) βˆ’ (𝑋 , π‘Œ))) < 𝑅)
 
Theoremipcnlem1 24994* The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    , = (Β·π‘–β€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &   π‘‡ = ((𝑅 / 2) / ((π‘β€˜π΄) + 1))    &   π‘ˆ = ((𝑅 / 2) / ((π‘β€˜π΅) + 𝑇))    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ+ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (((𝐴𝐷π‘₯) < π‘Ÿ ∧ (𝐡𝐷𝑦) < π‘Ÿ) β†’ (absβ€˜((𝐴 , 𝐡) βˆ’ (π‘₯ , 𝑦))) < 𝑅))
 
Theoremipcn 24995 The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
, = (Β·ifβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    &   πΎ = (TopOpenβ€˜β„‚fld)    β‡’   (π‘Š ∈ β„‚PreHil β†’ , ∈ ((𝐽 Γ—t 𝐽) Cn 𝐾))
 
Theoremcnmpt1ip 24996* Continuity of inner product; analogue of cnmpt12f 23391 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpenβ€˜π‘Š)    &   πΆ = (TopOpenβ€˜β„‚fld)    &    , = (Β·π‘–β€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐾 Cn 𝐽))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 , 𝐡)) ∈ (𝐾 Cn 𝐢))
 
Theoremcnmpt2ip 24997* Continuity of inner product; analogue of cnmpt22f 23400 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpenβ€˜π‘Š)    &   πΆ = (TopOpenβ€˜β„‚fld)    &    , = (Β·π‘–β€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ β„‚PreHil)    &   (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘Œ))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐿) Cn 𝐽))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐾 Γ—t 𝐿) Cn 𝐽))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (𝐴 , 𝐡)) ∈ ((𝐾 Γ—t 𝐿) Cn 𝐢))
 
Theoremcsscld 24998 A "closed subspace" in a subcomplex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐢 = (ClSubSpβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝑆 ∈ 𝐢) β†’ 𝑆 ∈ (Clsdβ€˜π½))
 
Theoremclsocv 24999 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘‚ = (ocvβ€˜π‘Š)    &   π½ = (TopOpenβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) = (π‘‚β€˜π‘†))
 
Theoremcphsscph 25000 A subspace of a subcomplex pre-Hilbert space is a subcomplex pre-Hilbert space. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 25-Sep-2022.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ β„‚PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ β„‚PreHil)
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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 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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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