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Theorem List for Metamath Proof Explorer - 28901-29000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnormsqi 28901 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 28902 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 28903 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))

Theoremnormsub0i 28904 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 28905 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 28906 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 28907 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 28908 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 28909 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 28910 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 28911 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 28912 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 28913 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 28914 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 28915 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 28916 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 28917 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 28918 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 28919 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 28920 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 28921 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 28922 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 28923 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 28924 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 28925 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 28926 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 28927 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 28853. (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 28928 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 28853. (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 28929 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
+ ∈ AbelOp

Theoremhilid 28930 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 28931 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 28932 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 28933 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 28934 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ NrmCVec

Theoremhhva 28935 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        + = ( +𝑣𝑈)

Theoremhhba 28936 The base set of Hilbert space. This theorem provides an independent proof of df-hba 28738 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        ℋ = (BaseSet‘𝑈)

Theoremhh0v 28937 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 28938 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        · = ( ·𝑠OLD𝑈)

Theoremhhvs 28939 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm        = ( −𝑣𝑈)

Theoremhhnm 28940 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       norm = (normCV𝑈)

Theoremhhims 28941 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (norm ∘ − )       𝐷 = (IndMet‘𝑈)

Theoremhhims2 28942 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐷 = (IndMet‘𝑈)       𝐷 = (norm ∘ − )

Theoremhhmet 28943 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 28944 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 28945 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 28946 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 28947 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 28948 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 28949 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 28950 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 28951 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28949. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (norm𝐵))

Theorembcs3 28952 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28949. (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 28953* 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 28954 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 28955* 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 28956* 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 28957* 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 28958 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 28959 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 28960* 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 28961* 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 28962* 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 28963 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)
𝐷 = (norm ∘ − )       𝐷 ∈ (Met‘ ℋ)

Theoremhilxmet 28964 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 28965 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 28966 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 28967 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 28968 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 28969* Lemma used for derivation of the completeness axiom ax-hcompl 28971 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 28970* Lemma used for derivation of the completeness axiom ax-hcompl 28971 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 28768; the 6th would be satisfied by eqid 2819; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 28684. (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 28971* 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 28972 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 28973 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm       𝑈 ∈ CHilOLD

Theoremhilcms 28974 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 28975 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 28976 Define the set of subspaces of a Hilbert space. See issh 28977 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 28977 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 28978* 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 28979 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 28980 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 28981 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 28982 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 28983 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 28984 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 28985 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 28986 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 28987 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 28988* 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 28989 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 28990 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 28991. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28992 and isch3 29010. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)
C = {S ∣ ( ⇝𝑣 “ (m ℕ)) ⊆ }

Theoremisch 28991 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 28992* 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 28993 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 28994 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 28995 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 28996 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐻C       𝐻S

Theoremch0 28997 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 28998 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 28999 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 29000 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

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