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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhhmet 28001 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 28002 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 28003 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 28004 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 28005 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 28006 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 28007 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 28008 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 28009 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28007. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (norm𝐵))

Theorembcs3 28010 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28007. (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 28011* 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 28012 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 28013* 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 28014* 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 28015* 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 28016 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 28017 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 28018* 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 28019* 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 28020* 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 28021 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (Met‘ ℋ)

Theoremhilxmet 28022 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 28023 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 28024 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 28025 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 28026 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 28027* Lemma used for derivation of the completeness axiom ax-hcompl 28029 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 28028* Lemma used for derivation of the completeness axiom ax-hcompl 28029 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 27826; the 6th would be satisfied by eqid 2620; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 27741. (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 28029* 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 28030 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 28031 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ CHilOLD

Theoremhilcms 28032 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 28033 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 28034 Define the set of subspaces of a Hilbert space. See issh 28035 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 28035 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 28036* 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 28037 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 28038 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 28039 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 28040 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 28041 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 28042 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 28043 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 28044 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 28045 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 28046* 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 28047 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 28048 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 28049. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28050 and isch3 28068. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
C = {S ∣ ( ⇝𝑣 “ (𝑚 ℕ)) ⊆ }

Theoremisch 28049 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 28050* 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 28051 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 28052 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 28053 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 28054 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻S

Theoremch0 28055 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 28056 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 28057 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 28058 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 28059 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 28060 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 28061 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 28062 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 28063 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 28064 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑣 :dom ⇝𝑣 ⟶ ℋ

Theoremhlimuni 28065 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 28066* 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 28067* 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 28068* 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 28069* 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 28070 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 28071 Prove if(𝐴C , 𝐴, ℋ) ∈ C (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
if(𝐴C , 𝐴, ℋ) ∈ C

Theoremhelsh 28072 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
ℋ ∈ S

Theoremshsspwh 28073 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
S ⊆ 𝒫 ℋ

Theoremchsspwh 28074 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
C ⊆ 𝒫 ℋ

Theoremhsn0elch 28075 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 28076 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 28077* 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 28078 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 28079* 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 28109 and chocvali 28128 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 28080 Define the zero for closed subspaces of Hilbert space. See h0elch 28082 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0 = {0}

Theoremelch0 28081 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
(𝐴 ∈ 0𝐴 = 0)

Theoremh0elch 28082 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 28083 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
0S

Theoremhhssva 28084 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 28085 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 28086 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 28087* Lemma for hhssabloilem 28088. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))       ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

Theoremhhssabloilem 28088 Lemma for hhssabloi 28089. Formerly part of proof for hhssabloi 28089 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 28089 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 28090 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
(𝐻S → ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp)

Theoremhhssnv 28091 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 28092 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)

Theoremhhsst 28093 A member of S is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S𝑊 ∈ (SubSp‘𝑈))

Theoremhhshsslem1 28094 Lemma for hhsssh 28096. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝑊 ∈ (SubSp‘𝑈)    &   𝐻 ⊆ ℋ       𝐻 = (BaseSet‘𝑊)

Theoremhhshsslem2 28095 Lemma for hhsssh 28096. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝑊 ∈ (SubSp‘𝑈)    &   𝐻 ⊆ ℋ       𝐻S

Theoremhhsssh 28096 The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

Theoremhhsssh2 28097 The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩       (𝐻S ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ))

Theoremhhssba 28098 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝐻 = (BaseSet‘𝑊)

Theoremhhssvs 28099 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       ( − ↾ (𝐻 × 𝐻)) = ( −𝑣𝑊)

Theoremhhssvsf 28100 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       ( − ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻

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