Home Metamath Proof ExplorerTheorem List (p. 282 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 - 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

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

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 >