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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsh0 28201 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 28202 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 28203 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 28204* 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 28205 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 28206 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 28207. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28208 and isch3 28226. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
C = {S ∣ ( ⇝𝑣 “ (𝑚 ℕ)) ⊆ }

Theoremisch 28207 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 28208* 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 28209 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 28210 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 28211 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 28212 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻S

Theoremch0 28213 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 28214 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 28215 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 28216 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 28217 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 28218 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 28219 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 28220 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 28221 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 28222 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑣 :dom ⇝𝑣 ⟶ ℋ

Theoremhlimuni 28223 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 28224* 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 28225* 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 28226* 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 28227* 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 28228 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 28229 Prove if(𝐴C , 𝐴, ℋ) ∈ C. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.)
if(𝐴C , 𝐴, ℋ) ∈ C

Theoremhelsh 28230 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
ℋ ∈ S

Theoremshsspwh 28231 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
S ⊆ 𝒫 ℋ

Theoremchsspwh 28232 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
C ⊆ 𝒫 ℋ

Theoremhsn0elch 28233 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 28234 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 28235* 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 28236 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 28237* 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 28267 and chocvali 28286 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 28238 Define the zero for closed subspaces of Hilbert space. See h0elch 28240 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
0 = {0}

Theoremelch0 28239 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
(𝐴 ∈ 0𝐴 = 0)

Theoremh0elch 28240 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 28241 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
0S

Theoremhhssva 28242 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 28243 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 28244 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 28245* Lemma for hhssabloilem 28246. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))       ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

Theoremhhssabloilem 28246 Lemma for hhssabloi 28247. Formerly part of proof for hhssabloi 28247 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 28247 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 28248 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)
(𝐻S → ( + ↾ (𝐻 × 𝐻)) ∈ AbelOp)

Theoremhhssnv 28249 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 28250 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 28251 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 28252 Lemma for hhsssh 28254. (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 28253 Lemma for hhsssh 28254. (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 28254 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 28255 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 28256 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 28257 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 28258 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

Theoremhhssph 28259 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S       𝑊 ∈ CPreHilOLD

Theoremhhssims 28260 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻S    &   𝐷 = ((norm ∘ − ) ↾ (𝐻 × 𝐻))       𝐷 = (IndMet‘𝑊)

Theoremhhssims2 28261 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       𝐷 = ((norm ∘ − ) ↾ (𝐻 × 𝐻))

Theoremhhssmet 28262 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       𝐷 ∈ (Met‘𝐻)

Theoremhhssmetdval 28263 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻S       ((𝐴𝐻𝐵𝐻) → (𝐴𝐷𝐵) = (norm‘(𝐴 𝐵)))

Theoremhhsscms 28264 The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐷 = (IndMet‘𝑊)    &   𝐻C       𝐷 ∈ (CMet‘𝐻)

Theoremhhssbn 28265 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻C       𝑊 ∈ CBan

Theoremhhsshl 28266 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩    &   𝐻C       𝑊 ∈ CHilOLD

Theoremocval 28267* Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐻 (𝑥 ·ih 𝑦) = 0})

Theoremocel 28268* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))

Theoremshocel 28269* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)
(𝐻S → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥𝐻 (𝐴 ·ih 𝑥) = 0)))

Theoremocsh 28270 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )

Theoremshocsh 28271 The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ∈ S )

Theoremocss 28272 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)

Theoremshocss 28273 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ⊆ ℋ)

Theoremoccon 28274 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴)))

Theoremoccon2 28275 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))))

Theoremoccon2i 28276 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
𝐴 ⊆ ℋ    &   𝐵 ⊆ ℋ       (𝐴𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))

Theoremoc0 28277 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐻S → 0 ∈ (⊥‘𝐻))

Theoremocorth 28278 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐻 ⊆ ℋ → ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))

Theoremshocorth 28279 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐻S → ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0))

Theoremococss 28280 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))

Theoremshococss 28281 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
(𝐴S𝐴 ⊆ (⊥‘(⊥‘𝐴)))

Theoremshorth 28282 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
(𝐻S → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴𝐺𝐵𝐻) → (𝐴 ·ih 𝐵) = 0)))

Theoremocin 28283 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
(𝐴S → (𝐴 ∩ (⊥‘𝐴)) = 0)

Theoremoccon3 28284 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)
((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)))

Theoremocnel 28285 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
((𝐻S𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0) → ¬ 𝐴𝐻)

Theoremchocvali 28286* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of 𝐴 is the set of vectors that are orthogonal to all vectors in 𝐴. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0}

Theoremshuni 28287 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(𝜑𝐻S )    &   (𝜑𝐾S )    &   (𝜑 → (𝐻𝐾) = 0)    &   (𝜑𝐴𝐻)    &   (𝜑𝐵𝐾)    &   (𝜑𝐶𝐻)    &   (𝜑𝐷𝐾)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremchocunii 28288 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐻C       (((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) ∧ (𝐶𝐻𝐷 ∈ (⊥‘𝐻))) → ((𝑅 = (𝐴 + 𝐵) ∧ 𝑅 = (𝐶 + 𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempjhthmo 28289* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴S𝐵S ∧ (𝐴𝐵) = 0) → ∃*𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝐶 = (𝑥 + 𝑦)))

Theoremoccllem 28290 Lemma for occl 28291. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝜑𝐴 ⊆ ℋ)    &   (𝜑𝐹 ∈ Cauchy)    &   (𝜑𝐹:ℕ⟶(⊥‘𝐴))    &   (𝜑𝐵𝐴)       (𝜑 → (( ⇝𝑣𝐹) ·ih 𝐵) = 0)

Theoremoccl 28291 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
(𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ C )

Theoremshoccl 28292 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴S → (⊥‘𝐴) ∈ C )

Theoremchoccl 28293 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)
(𝐴C → (⊥‘𝐴) ∈ C )

Theoremchoccli 28294 Closure of C orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴C       (⊥‘𝐴) ∈ C

19.4.4  Subspace sum, span, lattice join, lattice supremum

Definitiondf-shs 28295* Define subspace sum in S. See shsval 28299, shsval2i 28374, and shsval3i 28375 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
+ = (𝑥S , 𝑦S ↦ ( + “ (𝑥 × 𝑦)))

Definitiondf-span 28296* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 28320 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)
span = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦S𝑥𝑦})

Definitiondf-chj 28297* Define Hilbert lattice join. See chjval 28339 for its value and chjcl 28344 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to C; see sshjcl 28342. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))

Definitiondf-chsup 28298 Define the supremum of a set of Hilbert lattice elements. See chsupval2 28397 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice C, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 28326. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)
= (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘ 𝑥)))

Theoremshsval 28299 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) = ( + “ (𝐴 × 𝐵)))

Theoremshsss 28300 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
((𝐴S𝐵S ) → (𝐴 + 𝐵) ⊆ ℋ)

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