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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhhva 29101 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        + = ( +𝑣𝑈)
 
Theoremhhba 29102 The base set of Hilbert space. This theorem provides an independent proof of df-hba 28904 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        ℋ = (BaseSet‘𝑈)
 
Theoremhh0v 29103 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 29104 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        · = ( ·𝑠OLD𝑈)
 
Theoremhhvs 29105 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        = ( −𝑣𝑈)
 
Theoremhhnm 29106 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       norm = (normCV𝑈)
 
Theoremhhims 29107 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (norm ∘ − )       𝐷 = (IndMet‘𝑈)
 
Theoremhhims2 29108 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 = (norm ∘ − )
 
Theoremhhmet 29109 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 29110 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 29111 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 29112 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 29113 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 29114 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 29115 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 29116 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 29117 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 29115. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (norm𝐵))
 
Theorembcs3 29118 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 29115. (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 29119* 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 29120 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 29121* 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 29122* 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 29123* 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 29124 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 29125 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 29126* 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 29127* 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 29128* 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 29129 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (Met‘ ℋ)
 
Theoremhilxmet 29130 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 29131 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 29132 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 29133 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 29134 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 29135* Lemma used for derivation of the completeness axiom ax-hcompl 29137 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 29136* Lemma used for derivation of the completeness axiom ax-hcompl 29137 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 28934; the 6th would be satisfied by eqid 2738; the 7th by a given fixed Hilbert space; and the last by Theorem hlcompl 28850. (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 29137* 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 29138 The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 ∈ (CMet‘ ℋ)
 
Theoremhhhl 29139 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ CHilOLD
 
Theoremhilcms 29140 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 29141 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 29142 Define the set of subspaces of a Hilbert space. See issh 29143 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 29143 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 29144* 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 29145 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 29146 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 29147 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 29148 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 29149 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 29150 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
 
Theoremsh0 29151 The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻S → 0𝐻)
 
Theoremshaddcl 29152 Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 + 𝐵) ∈ 𝐻)
 
Theoremshmulcl 29153 Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
((𝐻S𝐴 ∈ ℂ ∧ 𝐵𝐻) → (𝐴 · 𝐵) ∈ 𝐻)
 
Theoremissh3 29154* Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → (𝐻S ↔ (0𝐻 ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))))
 
Theoremshsubcl 29155 Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
((𝐻S𝐴𝐻𝐵𝐻) → (𝐴 𝐵) ∈ 𝐻)
 
19.4.2  Closed subspaces
 
Definitiondf-ch 29156 Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 29157. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 29158 and isch3 29176. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
C = {S ∣ ( ⇝𝑣 “ (m ℕ)) ⊆ }
 
Theoremisch 29157 Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻m ℕ)) ⊆ 𝐻))
 
Theoremisch2 29158* Closed subspace 𝐻 of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻C ↔ (𝐻S ∧ ∀𝑓𝑥((𝑓:ℕ⟶𝐻𝑓𝑣 𝑥) → 𝑥𝐻)))
 
Theoremchsh 29159 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻C𝐻S )
 
Theoremchsssh 29160 Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
CS
 
Theoremchex 29161 The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
C ∈ V
 
Theoremchshii 29162 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻S
 
Theoremch0 29163 The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
(𝐻C → 0𝐻)
 
Theoremchss 29164 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
(𝐻C𝐻 ⊆ ℋ)
 
Theoremchel 29165 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)
 
Theoremchssii 29166 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻 ⊆ ℋ
 
Theoremcheli 29167 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C       (𝐴𝐻𝐴 ∈ ℋ)
 
Theoremchelii 29168 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴𝐻       𝐴 ∈ ℋ
 
Theoremchlimi 29169 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ V       ((𝐻C𝐹:ℕ⟶𝐻𝐹𝑣 𝐴) → 𝐴𝐻)
 
Theoremhlim0 29170 The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(ℕ × {0}) ⇝𝑣 0
 
Theoremhlimcaui 29171 If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐹𝑣 𝐴𝐹 ∈ Cauchy)
 
Theoremhlimf 29172 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑣 :dom ⇝𝑣 ⟶ ℋ
 
Theoremhlimuni 29173 A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.)
((𝐹𝑣 𝐴𝐹𝑣 𝐵) → 𝐴 = 𝐵)
 
Theoremhlimreui 29174* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(∃𝑥𝐻 𝐹𝑣 𝑥 ↔ ∃!𝑥𝐻 𝐹𝑣 𝑥)
 
Theoremhlimeui 29175* The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(∃𝑥 𝐹𝑣 𝑥 ↔ ∃!𝑥 𝐹𝑣 𝑥)
 
Theoremisch3 29176* A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐻C ↔ (𝐻S ∧ ∀𝑓 ∈ Cauchy (𝑓:ℕ⟶𝐻 → ∃𝑥𝐻 𝑓𝑣 𝑥)))
 
Theoremchcompl 29177* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐹 ∈ Cauchy ∧ 𝐹:ℕ⟶𝐻) → ∃𝑥𝐻 𝐹𝑣 𝑥)
 
Theoremhelch 29178 The unit Hilbert lattice element (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.)
ℋ ∈ C
 
Theoremifchhv 29179 Prove if(𝐴C , 𝐴, ℋ) ∈ C. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
if(𝐴C , 𝐴, ℋ) ∈ C
 
Theoremhelsh 29180 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
ℋ ∈ S
 
Theoremshsspwh 29181 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
S ⊆ 𝒫 ℋ
 
Theoremchsspwh 29182 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
C ⊆ 𝒫 ℋ
 
Theoremhsn0elch 29183 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
{0} ∈ C
 
Theoremnorm1 29184 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
 
Theoremnorm1exi 29185* A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
𝐻S       (∃𝑥𝐻 𝑥 ≠ 0 ↔ ∃𝑦𝐻 (norm𝑦) = 1)
 
Theoremnorm1hex 29186 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
(∃𝑥 ∈ ℋ 𝑥 ≠ 0 ↔ ∃𝑦 ∈ ℋ (norm𝑦) = 1)
 
19.4.3  Orthocomplements
 
Definitiondf-oc 29187* Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 29215 and chocvali 29234 for its value. Textbooks usually denote this unary operation with the symbol as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧𝑥 (𝑦 ·ih 𝑧) = 0})
 
Definitiondf-ch0 29188 Define the zero for closed subspaces of Hilbert space. See h0elch 29190 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0 = {0}
 
Theoremelch0 29189 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
(𝐴 ∈ 0𝐴 = 0)
 
Theoremh0elch 29190 The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0C
 
Theoremh0elsh 29191 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
0S
 
Theoremhhssva 29192 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       ( + ↾ (𝐻 × 𝐻)) = ( +𝑣𝑊)
 
Theoremhhsssm 29193 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       ( · ↾ (ℂ × 𝐻)) = ( ·𝑠OLD𝑊)
 
Theoremhhssnm 29194 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (norm𝐻) = (normCV𝑊)
 
Theoremissubgoilem 29195* Lemma for hhssabloilem 29196. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))       ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))
 
Theoremhhssabloilem 29196 Lemma for hhssabloi 29197. Formerly part of proof for hhssabloi 29197 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.)
𝐻S       ( + ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( + ↾ (𝐻 × 𝐻)) ⊆ + )
 
Theoremhhssabloi 29197 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Proof shortened by AV, 27-Aug-2021.) (New usage is discouraged.)
𝐻S       ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp
 
Theoremhhssablo 29198 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
(𝐻S → ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp)
 
Theoremhhssnv 29199 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝑊 ∈ NrmCVec
 
Theoremhhssnvt 29200 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S𝑊 ∈ NrmCVec)
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