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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | normne0 28901 | A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ)) | ||
Theorem | normcli 28902 | Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ (normℎ‘𝐴) ∈ ℝ | ||
Theorem | normsqi 28903 | The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴) | ||
Theorem | norm-i-i 28904 | Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ ⇒ ⊢ ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ) | ||
Theorem | normsq 28905 | The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴)) | ||
Theorem | normsub0i 28906 | 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 ⇒ ⊢ ((normℎ‘(𝐴 −ℎ 𝐵)) = 0 ↔ 𝐴 = 𝐵) | ||
Theorem | normsub0 28907 | 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) → ((normℎ‘(𝐴 −ℎ 𝐵)) = 0 ↔ 𝐴 = 𝐵)) | ||
Theorem | norm-ii-i 28908 | 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 ⇒ ⊢ (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵)) | ||
Theorem | norm-ii 28909 | 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) → (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵))) | ||
Theorem | norm-iii-i 28910 | Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)) | ||
Theorem | norm-iii 28911 | Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵))) | ||
Theorem | normsubi 28912 | 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 ⇒ ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴)) | ||
Theorem | normpythi 28913 | 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 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) | ||
Theorem | normsub 28914 | 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) → (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴))) | ||
Theorem | normneg 28915 | The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (normℎ‘(-1 ·ℎ 𝐴)) = (normℎ‘𝐴)) | ||
Theorem | normpyth 28916 | 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 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))) | ||
Theorem | normpyc 28917 | 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 → (normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵)))) | ||
Theorem | norm3difi 28918 | 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 ⇒ ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) | ||
Theorem | norm3adifii 28919 | 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‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) | ||
Theorem | norm3lem 28920 | 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 ⇒ ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷) | ||
Theorem | norm3dif 28921 | 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) → (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵)))) | ||
Theorem | norm3dif2 28922 | Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐶 −ℎ 𝐴)) + (normℎ‘(𝐶 −ℎ 𝐵)))) | ||
Theorem | norm3lemt 28923 | 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)) → (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷)) | ||
Theorem | norm3adifi 28924 | 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‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵))) | ||
Theorem | normpari 28925 | 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 ⇒ ⊢ (((normℎ‘(𝐴 −ℎ 𝐵))↑2) + ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) = ((2 · ((normℎ‘𝐴)↑2)) + (2 · ((normℎ‘𝐵)↑2))) | ||
Theorem | normpar 28926 | 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) → (((normℎ‘(𝐴 −ℎ 𝐵))↑2) + ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) = ((2 · ((normℎ‘𝐴)↑2)) + (2 · ((normℎ‘𝐵)↑2)))) | ||
Theorem | normpar2i 28927 | 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 ⇒ ⊢ ((normℎ‘(𝐴 −ℎ 𝐵))↑2) = (((2 · ((normℎ‘(𝐴 −ℎ 𝐶))↑2)) + (2 · ((normℎ‘(𝐵 −ℎ 𝐶))↑2))) − ((normℎ‘((𝐴 +ℎ 𝐵) −ℎ (2 ·ℎ 𝐶)))↑2)) | ||
Theorem | polid2i 28928 | 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 +ℎ 𝐶) ·ih (𝐷 +ℎ 𝐵)) − ((𝐴 −ℎ 𝐶) ·ih (𝐷 −ℎ 𝐵))) + (i · (((𝐴 +ℎ (i ·ℎ 𝐶)) ·ih (𝐷 +ℎ (i ·ℎ 𝐵))) − ((𝐴 −ℎ (i ·ℎ 𝐶)) ·ih (𝐷 −ℎ (i ·ℎ 𝐵)))))) / 4) | ||
Theorem | polidi 28929 | 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 28855. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·ih 𝐵) = (((((normℎ‘(𝐴 +ℎ 𝐵))↑2) − ((normℎ‘(𝐴 −ℎ 𝐵))↑2)) + (i · (((normℎ‘(𝐴 +ℎ (i ·ℎ 𝐵)))↑2) − ((normℎ‘(𝐴 −ℎ (i ·ℎ 𝐵)))↑2)))) / 4) | ||
Theorem | polid 28930 | 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 28855. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (((((normℎ‘(𝐴 +ℎ 𝐵))↑2) − ((normℎ‘(𝐴 −ℎ 𝐵))↑2)) + (i · (((normℎ‘(𝐴 +ℎ (i ·ℎ 𝐵)))↑2) − ((normℎ‘(𝐴 −ℎ (i ·ℎ 𝐵)))↑2)))) / 4)) | ||
Theorem | hilablo 28931 | Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
⊢ +ℎ ∈ AbelOp | ||
Theorem | hilid 28932 | The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.) |
⊢ (GId‘ +ℎ ) = 0ℎ | ||
Theorem | hilvc 28933 | Hilbert space is a complex vector space. Vector addition is +ℎ, and scalar product is ·ℎ. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
⊢ 〈 +ℎ , ·ℎ 〉 ∈ CVecOLD | ||
Theorem | hilnormi 28934 | 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 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
Theorem | hilhhi 28935 | Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | ||
Theorem | hhnv 28936 | Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ 𝑈 ∈ NrmCVec | ||
Theorem | hhva 28937 | The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ +ℎ = ( +𝑣 ‘𝑈) | ||
Theorem | hhba 28938 | The base set of Hilbert space. This theorem provides an independent proof of df-hba 28740 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ ℋ = (BaseSet‘𝑈) | ||
Theorem | hh0v 28939 | The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ 0ℎ = (0vec‘𝑈) | ||
Theorem | hhsm 28940 | The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | ||
Theorem | hhvs 28941 | The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ −ℎ = ( −𝑣 ‘𝑈) | ||
Theorem | hhnm 28942 | The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
Theorem | hhims 28943 | The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 = (IndMet‘𝑈) | ||
Theorem | hhims2 28944 | Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 = (normℎ ∘ −ℎ ) | ||
Theorem | hhmet 28945 | The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ) | ||
Theorem | hhxmet 28946 | The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ) | ||
Theorem | hhmetdval 28947 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
Theorem | hhip 28948 | 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ ·ih = (·𝑖OLD‘𝑈) | ||
Theorem | hhph 28949 | The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ 𝑈 ∈ CPreHilOLD | ||
Theorem | bcsiALT 28950 | 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)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵)) | ||
Theorem | bcsiHIL 28951 | 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)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵)) | ||
Theorem | bcs 28952 | 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)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵))) | ||
Theorem | bcs2 28953 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28951. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (normℎ‘𝐵)) | ||
Theorem | bcs3 28954 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 28951. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (normℎ‘𝐵) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (normℎ‘𝐴)) | ||
Theorem | hcau 28955* | 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 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) | ||
Theorem | hcauseq 28956 | 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) | ||
Theorem | hcaucvg 28957* | 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 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴) | ||
Theorem | seq1hcau 28958* | 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 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) | ||
Theorem | hlimi 28959* | 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 ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) | ||
Theorem | hlimseqi 28960 | A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹:ℕ⟶ ℋ) | ||
Theorem | hlimveci 28961 | Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ) | ||
Theorem | hlimconvi 28962* | 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 ⇒ ⊢ ((𝐹 ⇝𝑣 𝐴 ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝐵) | ||
Theorem | hlim2 28963* | 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) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) | ||
Theorem | hlimadd 28964* | Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) & ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) ⇒ ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) | ||
Theorem | hilmet 28965 | The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ) | ||
Theorem | hilxmet 28966 | The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ) | ||
Theorem | hilmetdval 28967 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
Theorem | hilims 28968 | Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.) |
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝐷 = (normℎ ∘ −ℎ ) | ||
Theorem | hhcau 28969 | The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) | ||
Theorem | hhlm 28970 | The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ⇝𝑣 = ((⇝𝑡‘𝐽) ↾ ( ℋ ↑m ℕ)) | ||
Theorem | hhcmpl 28971* | Lemma used for derivation of the completeness axiom ax-hcompl 28973 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Theorem | hilcompl 28972* | Lemma used for derivation of the completeness axiom ax-hcompl 28973 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 28770; the 6th would be satisfied by eqid 2821; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 28686. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CHilOLD & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Axiom | ax-hcompl 28973* | Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Theorem | hhcms 28974 | 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ) | ||
Theorem | hhhl 28975 | The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ⇒ ⊢ 𝑈 ∈ CHilOLD | ||
Theorem | hilcms 28976 | The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ) | ||
Theorem | hilhl 28977 | The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.) |
⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ CHilOLD | ||
Definition | df-sh 28978 | Define the set of subspaces of a Hilbert space. See issh 28979 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.) |
⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)} | ||
Theorem | issh 28979 | 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.) |
⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | ||
Theorem | issh2 28980* | 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.) |
⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))) | ||
Theorem | shss 28981 | 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.) |
⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | ||
Theorem | shel 28982 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | ||
Theorem | shex 28983 | The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
⊢ Sℋ ∈ V | ||
Theorem | shssii 28984 | A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝐻 ⊆ ℋ | ||
Theorem | sheli 28985 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Sℋ ⇒ ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) | ||
Theorem | shelii 28986 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Sℋ & ⊢ 𝐴 ∈ 𝐻 ⇒ ⊢ 𝐴 ∈ ℋ | ||
Theorem | sh0 28987 | 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.) |
⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) | ||
Theorem | shaddcl 28988 | Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻) | ||
Theorem | shmulcl 28989 | Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻) | ||
Theorem | issh3 28990* | Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ Sℋ ↔ (0ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))) | ||
Theorem | shsubcl 28991 | Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | ||
Definition | df-ch 28992 | 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 28993. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28994 and isch3 29012. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | ||
Theorem | isch 28993 | Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | ||
Theorem | isch2 28994* | 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.) |
⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) | ||
Theorem | chsh 28995 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | ||
Theorem | chsssh 28996 | 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.) |
⊢ Cℋ ⊆ Sℋ | ||
Theorem | chex 28997 | The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
⊢ Cℋ ∈ V | ||
Theorem | chshii 28998 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
Theorem | ch0 28999 | The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻) | ||
Theorem | chss 29000 | A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) |
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