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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hvsubcan 31001 | Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = (𝐴 −ℎ 𝐶) ↔ 𝐵 = 𝐶)) | ||
| Theorem | hvsubcan2 31002 | Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐶) = (𝐵 −ℎ 𝐶) ↔ 𝐴 = 𝐵)) | ||
| Theorem | hvsub0 31003 | Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 −ℎ 0ℎ) = 𝐴) | ||
| Theorem | hvsubadd 31004 | Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴)) | ||
| Theorem | hvaddsub4 31005 | Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) = (𝐶 +ℎ 𝐷) ↔ (𝐴 −ℎ 𝐶) = (𝐷 −ℎ 𝐵))) | ||
| Axiom | ax-hfi 31006 | Inner product maps pairs from ℋ to ℂ. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ ·ih :( ℋ × ℋ)⟶ℂ | ||
| Theorem | hicl 31007 | Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ) | ||
| Theorem | hicli 31008 | Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·ih 𝐵) ∈ ℂ | ||
| Axiom | ax-his1 31009 | Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15119 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7403 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4608. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31777. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) | ||
| Axiom | ax-his2 31010 | Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))) | ||
| Axiom | ax-his3 31011 | Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐵 ·ih (𝐴 ·ℎ 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 31777 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))) | ||
| Axiom | ax-his4 31012 | Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | ||
| Theorem | his5 31013 | Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶))) | ||
| Theorem | his52 31014 | Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih ((∗‘𝐴) ·ℎ 𝐶)) = (𝐴 · (𝐵 ·ih 𝐶))) | ||
| Theorem | his35 31015 | Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷))) | ||
| Theorem | his35i 31016 | Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷)) | ||
| Theorem | his7 31017 | Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶))) | ||
| Theorem | hiassdi 31018 | Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷))) | ||
| Theorem | his2sub 31019 | Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) − (𝐵 ·ih 𝐶))) | ||
| Theorem | his2sub2 31020 | Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 −ℎ 𝐶)) = ((𝐴 ·ih 𝐵) − (𝐴 ·ih 𝐶))) | ||
| Theorem | hire 31021 | A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) ∈ ℝ ↔ (𝐴 ·ih 𝐵) = (𝐵 ·ih 𝐴))) | ||
| Theorem | hiidrcl 31022 | Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ) | ||
| Theorem | hi01 31023 | Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | ||
| Theorem | hi02 31024 | Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = 0) | ||
| Theorem | hiidge0 31025 | Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴)) | ||
| Theorem | his6 31026 | Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → ((𝐴 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ)) | ||
| Theorem | his1i 31027 | Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)) | ||
| Theorem | abshicom 31028 | Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) = (abs‘(𝐵 ·ih 𝐴))) | ||
| Theorem | hial0 31029* | A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = 0 ↔ 𝐴 = 0ℎ)) | ||
| Theorem | hial02 31030* | A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ)) | ||
| Theorem | hisubcomi 31031 | Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶)) | ||
| Theorem | hi2eq 31032 | Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵)) | ||
| Theorem | hial2eq 31033* | Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵)) | ||
| Theorem | hial2eq2 31034* | Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = (𝑥 ·ih 𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | orthcom 31035 | Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 ↔ (𝐵 ·ih 𝐴) = 0)) | ||
| Theorem | normlem0 31036 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ ⇒ ⊢ ((𝐹 −ℎ (𝑆 ·ℎ 𝐺)) ·ih (𝐹 −ℎ (𝑆 ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘𝑆) · (𝐹 ·ih 𝐺))) + ((-𝑆 · (𝐺 ·ih 𝐹)) + ((𝑆 · (∗‘𝑆)) · (𝐺 ·ih 𝐺)))) | ||
| Theorem | normlem1 31037 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝑅 ∈ ℝ & ⊢ (abs‘𝑆) = 1 ⇒ ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) | ||
| Theorem | normlem2 31038 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ⇒ ⊢ 𝐵 ∈ ℝ | ||
| Theorem | normlem3 31039 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) & ⊢ 𝐴 = (𝐺 ·ih 𝐺) & ⊢ 𝐶 = (𝐹 ·ih 𝐹) & ⊢ 𝑅 ∈ ℝ ⇒ ⊢ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) | ||
| Theorem | normlem4 31040 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) & ⊢ 𝐴 = (𝐺 ·ih 𝐺) & ⊢ 𝐶 = (𝐹 ·ih 𝐹) & ⊢ 𝑅 ∈ ℝ & ⊢ (abs‘𝑆) = 1 ⇒ ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) | ||
| Theorem | normlem5 31041 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) & ⊢ 𝐴 = (𝐺 ·ih 𝐺) & ⊢ 𝐶 = (𝐹 ·ih 𝐹) & ⊢ 𝑅 ∈ ℝ & ⊢ (abs‘𝑆) = 1 ⇒ ⊢ 0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) | ||
| Theorem | normlem6 31042 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) & ⊢ 𝐴 = (𝐺 ·ih 𝐺) & ⊢ 𝐶 = (𝐹 ·ih 𝐹) & ⊢ (abs‘𝑆) = 1 ⇒ ⊢ (abs‘𝐵) ≤ (2 · ((√‘𝐴) · (√‘𝐶))) | ||
| Theorem | normlem7 31043 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝑆 ∈ ℂ & ⊢ 𝐹 ∈ ℋ & ⊢ 𝐺 ∈ ℋ & ⊢ (abs‘𝑆) = 1 ⇒ ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) | ||
| Theorem | normlem8 31044 | Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶))) | ||
| Theorem | normlem9 31045 | Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ & ⊢ 𝐶 ∈ ℋ & ⊢ 𝐷 ∈ ℋ ⇒ ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) − ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶))) | ||
| Theorem | normlem7tALT 31046 | Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ ((𝑆 ∈ ℂ ∧ (abs‘𝑆) = 1) → (((∗‘𝑆) · (𝐴 ·ih 𝐵)) + (𝑆 · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))))) | ||
| Theorem | bcseqi 31047 | Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 31107. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (((𝐴 ·ih 𝐵) · (𝐵 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) · (𝐵 ·ih 𝐵)) ↔ ((𝐵 ·ih 𝐵) ·ℎ 𝐴) = ((𝐴 ·ih 𝐵) ·ℎ 𝐵)) | ||
| Theorem | normlem9at 31048 | Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))) | ||
| Theorem | dfhnorm2 31049 | Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
| ⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥))) | ||
| Theorem | normf 31050 | The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
| ⊢ normℎ: ℋ⟶ℝ | ||
| Theorem | normval 31051 | The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴))) | ||
| Theorem | normcl 31052 | Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | ||
| Theorem | normge0 31053 | The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) | ||
| Theorem | normgt0 31054 | The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴))) | ||
| Theorem | norm0 31055 | The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ (normℎ‘0ℎ) = 0 | ||
| Theorem | norm-i 31056 | Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ)) | ||
| Theorem | normne0 31057 | 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 31058 | 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 31059 | 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 31060 | 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 31061 | 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 31062 | 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 31063 | 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 31064 | 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 31065 | 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 31066 | 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 31067 | 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 31068 | 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 31069 | 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 31070 | 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 31071 | 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 31072 | 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 31073 | 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 31074 | 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 31075 | 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 31076 | 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 31077 | 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 31078 | 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 31079 | 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 31080 | 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 31081 | 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 31082 | 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 31083 | 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 31084 | 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 31085 | 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 31011. (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 31086 | 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 31011. (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 31087 | Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
| ⊢ +ℎ ∈ AbelOp | ||
| Theorem | hilid 31088 | 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 31089 | 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 31090 | 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 31091 | 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 31092 | 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 31093 | 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 31094 | The base set of Hilbert space. This theorem provides an independent proof of df-hba 30896 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ ℋ = (BaseSet‘𝑈) | ||
| Theorem | hh0v 31095 | 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 31096 | 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 31097 | 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 31098 | 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 31099 | 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 31100 | Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 = (normℎ ∘ −ℎ ) | ||
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