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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnormlem7 28101 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ ℂ    &   𝐹 ∈ ℋ    &   𝐺 ∈ ℋ    &   (abs‘𝑆) = 1       (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))))
 
Theoremnormlem8 28102 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 + 𝐵) ·ih (𝐶 + 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
 
Theoremnormlem9 28103 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       ((𝐴 𝐵) ·ih (𝐶 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) − ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
 
Theoremnormlem7tALT 28104 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝑆 ∈ ℂ ∧ (abs‘𝑆) = 1) → (((∗‘𝑆) · (𝐴 ·ih 𝐵)) + (𝑆 · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
 
Theorembcseqi 28105 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 28165. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((𝐴 ·ih 𝐵) · (𝐵 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) · (𝐵 ·ih 𝐵)) ↔ ((𝐵 ·ih 𝐵) · 𝐴) = ((𝐴 ·ih 𝐵) · 𝐵))
 
Theoremnormlem9at 28106 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))))
 
19.2.2  Norms
 
Theoremdfhnorm2 28107 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
norm = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
 
Theoremnormf 28108 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
norm: ℋ⟶ℝ
 
Theoremnormval 28109 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
 
Theoremnormcl 28110 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
 
Theoremnormge0 28111 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → 0 ≤ (norm𝐴))
 
Theoremnormgt0 28112 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
 
Theoremnorm0 28113 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(norm‘0) = 0
 
Theoremnorm-i 28114 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
 
Theoremnormne0 28115 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
 
Theoremnormcli 28116 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       (norm𝐴) ∈ ℝ
 
Theoremnormsqi 28117 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
 
Theoremnorm-i-i 28118 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ       ((norm𝐴) = 0 ↔ 𝐴 = 0)
 
Theoremnormsq 28119 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
 
Theoremnormsub0i 28120 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((norm‘(𝐴 𝐵)) = 0 ↔ 𝐴 = 𝐵)
 
Theoremnormsub0 28121 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((norm‘(𝐴 𝐵)) = 0 ↔ 𝐴 = 𝐵))
 
Theoremnorm-ii-i 28122 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (norm‘(𝐴 + 𝐵)) ≤ ((norm𝐴) + (norm𝐵))
 
Theoremnorm-ii 28123 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 + 𝐵)) ≤ ((norm𝐴) + (norm𝐵)))
 
Theoremnorm-iii-i 28124 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℋ       (norm‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (norm𝐵))
 
Theoremnorm-iii 28125 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (norm𝐵)))
 
Theoremnormsubi 28126 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (norm‘(𝐴 𝐵)) = (norm‘(𝐵 𝐴))
 
Theoremnormpythi 28127 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((𝐴 ·ih 𝐵) = 0 → ((norm‘(𝐴 + 𝐵))↑2) = (((norm𝐴)↑2) + ((norm𝐵)↑2)))
 
Theoremnormsub 28128 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (norm‘(𝐴 𝐵)) = (norm‘(𝐵 𝐴)))
 
Theoremnormneg 28129 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (norm‘(-1 · 𝐴)) = (norm𝐴))
 
Theoremnormpyth 28130 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((norm‘(𝐴 + 𝐵))↑2) = (((norm𝐴)↑2) + ((norm𝐵)↑2))))
 
Theoremnormpyc 28131 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → (norm𝐴) ≤ (norm‘(𝐴 + 𝐵))))
 
Theoremnorm3difi 28132 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐴 𝐶)) + (norm‘(𝐶 𝐵)))
 
Theoremnorm3adifii 28133 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       (abs‘((norm‘(𝐴 𝐶)) − (norm‘(𝐵 𝐶)))) ≤ (norm‘(𝐴 𝐵))
 
Theoremnorm3lem 28134 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℝ       (((norm‘(𝐴 𝐶)) < (𝐷 / 2) ∧ (norm‘(𝐶 𝐵)) < (𝐷 / 2)) → (norm‘(𝐴 𝐵)) < 𝐷)
 
Theoremnorm3dif 28135 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐴 𝐶)) + (norm‘(𝐶 𝐵))))
 
Theoremnorm3dif2 28136 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (norm‘(𝐴 𝐵)) ≤ ((norm‘(𝐶 𝐴)) + (norm‘(𝐶 𝐵))))
 
Theoremnorm3lemt 28137 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((norm‘(𝐴 𝐶)) < (𝐷 / 2) ∧ (norm‘(𝐶 𝐵)) < (𝐷 / 2)) → (norm‘(𝐴 𝐵)) < 𝐷))
 
Theoremnorm3adifi 28138 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
𝐶 ∈ ℋ       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((norm‘(𝐴 𝐶)) − (norm‘(𝐵 𝐶)))) ≤ (norm‘(𝐴 𝐵)))
 
Theoremnormpari 28139 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
 
Theoremnormpar 28140 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))))
 
Theoremnormpar2i 28141 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ       ((norm‘(𝐴 𝐵))↑2) = (((2 · ((norm‘(𝐴 𝐶))↑2)) + (2 · ((norm‘(𝐵 𝐶))↑2))) − ((norm‘((𝐴 + 𝐵) − (2 · 𝐶)))↑2))
 
Theorempolid2i 28142 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℋ    &   𝐷 ∈ ℋ       (𝐴 ·ih 𝐵) = (((((𝐴 + 𝐶) ·ih (𝐷 + 𝐵)) − ((𝐴 𝐶) ·ih (𝐷 𝐵))) + (i · (((𝐴 + (i · 𝐶)) ·ih (𝐷 + (i · 𝐵))) − ((𝐴 (i · 𝐶)) ·ih (𝐷 (i · 𝐵)))))) / 4)
 
Theorempolidi 28143 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 28069. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐴 ·ih 𝐵) = (((((norm‘(𝐴 + 𝐵))↑2) − ((norm‘(𝐴 𝐵))↑2)) + (i · (((norm‘(𝐴 + (i · 𝐵)))↑2) − ((norm‘(𝐴 (i · 𝐵)))↑2)))) / 4)
 
Theorempolid 28144 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 28069. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (((((norm‘(𝐴 + 𝐵))↑2) − ((norm‘(𝐴 𝐵))↑2)) + (i · (((norm‘(𝐴 + (i · 𝐵)))↑2) − ((norm‘(𝐴 (i · 𝐵)))↑2)))) / 4))
 
19.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 28145 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
+ ∈ AbelOp
 
Theoremhilid 28146 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
(GId‘ + ) = 0
 
Theoremhilvc 28147 Hilbert space is a complex vector space. Vector addition is +, and scalar product is ·. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⟨ + , · ⟩ ∈ CVecOLD
 
Theoremhilnormi 28148 Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝑈 ∈ NrmCVec       norm = (normCV𝑈)
 
Theoremhilhhi 28149 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    + = ( +𝑣𝑈)    &    · = ( ·𝑠OLD𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝑈 ∈ NrmCVec       𝑈 = ⟨⟨ + , · ⟩, norm
 
Theoremhhnv 28150 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ NrmCVec
 
Theoremhhva 28151 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        + = ( +𝑣𝑈)
 
Theoremhhba 28152 The base set of Hilbert space. This theorem provides an independent proof of df-hba 27954 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        ℋ = (BaseSet‘𝑈)
 
Theoremhh0v 28153 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       0 = (0vec𝑈)
 
Theoremhhsm 28154 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        · = ( ·𝑠OLD𝑈)
 
Theoremhhvs 28155 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        = ( −𝑣𝑈)
 
Theoremhhnm 28156 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       norm = (normCV𝑈)
 
Theoremhhims 28157 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (norm ∘ − )       𝐷 = (IndMet‘𝑈)
 
Theoremhhims2 28158 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 = (norm ∘ − )
 
Theoremhhmet 28159 The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 ∈ (Met‘ ℋ)
 
Theoremhhxmet 28160 The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 ∈ (∞Met‘ ℋ)
 
Theoremhhmetdval 28161 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))
 
Theoremhhip 28162 The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        ·ih = (·𝑖OLD𝑈)
 
Theoremhhph 28163 The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ CPreHilOLD
 
19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality
 
TheorembcsiALT 28164 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))
 
TheorembcsiHIL 28165 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))
 
Theorembcs 28166 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
 
Theorembcs2 28167 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28165. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (norm𝐵))
 
Theorembcs3 28168 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28165. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm𝐵) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (norm𝐴))
 
19.3  Cauchy sequences and completeness axiom
 
19.3.1  Cauchy sequences and limits
 
Theoremhcau 28169* Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑦) − (𝐹𝑧))) < 𝑥))
 
Theoremhcauseq 28170 A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐹 ∈ Cauchy → 𝐹:ℕ⟶ ℋ)
 
Theoremhcaucvg 28171* A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
((𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑦) − (𝐹𝑧))) < 𝐴)
 
Theoremseq1hcau 28172* A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐹:ℕ⟶ ℋ → (𝐹 ∈ Cauchy ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑦) − (𝐹𝑧))) < 𝑥))
 
Theoremhlimi 28173* Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝐴 ∈ V       (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
 
Theoremhlimseqi 28174 A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ V       (𝐹𝑣 𝐴𝐹:ℕ⟶ ℋ)
 
Theoremhlimveci 28175 Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ V       (𝐹𝑣 𝐴𝐴 ∈ ℋ)
 
Theoremhlimconvi 28176* Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝐴 ∈ V       ((𝐹𝑣 𝐴𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝐵)
 
Theoremhlim2 28177* The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
 
Theoremhlimadd 28178* Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝜑𝐹:ℕ⟶ ℋ)    &   (𝜑𝐺:ℕ⟶ ℋ)    &   (𝜑𝐹𝑣 𝐴)    &   (𝜑𝐺𝑣 𝐵)    &   𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹𝑛) + (𝐺𝑛)))       (𝜑𝐻𝑣 (𝐴 + 𝐵))
 
19.3.2  Derivation of the completeness axiom from ZF set theory
 
Theoremhilmet 28179 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (Met‘ ℋ)
 
Theoremhilxmet 28180 The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (∞Met‘ ℋ)
 
Theoremhilmetdval 28181 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))
 
Theoremhilims 28182 Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    + = ( +𝑣𝑈)    &    · = ( ·𝑠OLD𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝑈 ∈ NrmCVec       𝐷 = (norm ∘ − )
 
Theoremhhcau 28183 The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑𝑚 ℕ))
 
Theoremhhlm 28184 The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)       𝑣 = ((⇝𝑡𝐽) ↾ ( ℋ ↑𝑚 ℕ))
 
Theoremhhcmpl 28185* Lemma used for derivation of the completeness axiom ax-hcompl 28187 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡𝐽)𝑥)       (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)
 
Theoremhilcompl 28186* Lemma used for derivation of the completeness axiom ax-hcompl 28187 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 27984; the 6th would be satisfied by eqid 2651; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 27899. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
ℋ = (BaseSet‘𝑈)    &    + = ( +𝑣𝑈)    &    · = ( ·𝑠OLD𝑈)    &    ·ih = (·𝑖OLD𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝐽 = (MetOpen‘𝐷)    &   𝑈 ∈ CHilOLD    &   (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡𝐽)𝑥)       (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)
 
19.3.3  Completeness postulate for a Hilbert space
 
Axiomax-hcompl 28187* Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
(𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹𝑣 𝑥)
 
19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces
 
Theoremhhcms 28188 The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 ∈ (CMet‘ ℋ)
 
Theoremhhhl 28189 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ CHilOLD
 
Theoremhilcms 28190 The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (CMet‘ ℋ)
 
Theoremhilhl 28191 The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.)
⟨⟨ + , · ⟩, norm⟩ ∈ CHilOLD
 
19.4  Subspaces and projections
 
19.4.1  Subspaces
 
Definitiondf-sh 28192 Define the set of subspaces of a Hilbert space. See issh 28193 for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of [Beran] p. 95. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
 
Theoremissh 28193 Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
 
Theoremissh2 28194* Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
 
Theoremshss 28195 A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻S𝐻 ⊆ ℋ)
 
Theoremshel 28196 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)
 
Theoremshex 28197 The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
S ∈ V
 
Theoremshssii 28198 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻S       𝐻 ⊆ ℋ
 
Theoremsheli 28199 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻S       (𝐴𝐻𝐴 ∈ ℋ)
 
Theoremshelii 28200 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻S    &   𝐴𝐻       𝐴 ∈ ℋ
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