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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | nmosetn0 31301* | The set in the supremum of the operator norm definition df-nmoo 31281 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘(𝑇‘𝑍)) ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑋 ((𝑀‘𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑦)))}) | ||
| Theorem | nmoxr 31302 | The norm of an operator is an extended real. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) ∈ ℝ*) | ||
| Theorem | nmooge0 31303 | The norm of an operator is nonnegative. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → 0 ≤ (𝑁‘𝑇)) | ||
| Theorem | nmorepnf 31304 | The norm of an operator is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞)) | ||
| Theorem | nmoreltpnf 31305 | The norm of any operator is real iff it is less than plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞)) | ||
| Theorem | nmogtmnf 31306 | The norm of an operator is greater than minus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → -∞ < (𝑁‘𝑇)) | ||
| Theorem | nmoolb 31307 | A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝐿‘𝐴) ≤ 1)) → (𝑀‘(𝑇‘𝐴)) ≤ (𝑁‘𝑇)) | ||
| Theorem | nmoubi 31308* | An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ*) → ((𝑁‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑋 ((𝐿‘𝑥) ≤ 1 → (𝑀‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
| Theorem | nmoub3i 31309* | An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝐴 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ (abs‘𝐴)) | ||
| Theorem | nmoub2i 31310* | An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) → (𝑁‘𝑇) ≤ 𝐴) | ||
| Theorem | nmobndi 31311* | Two ways to express that an operator is bounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟))) | ||
| Theorem | nmounbi 31312* | Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))) | ||
| Theorem | nmounbseqi 31313* | An unbounded operator determines an unbounded sequence. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘)))))) | ||
| Theorem | nmounbseqiALT 31314* | Alternate shorter proof of nmounbseqi 31313 based on Axioms ax-reg 9564 and ax-ac2 10513 instead of ax-cc 10485. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ (𝑁‘𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓‘𝑘)))))) | ||
| Theorem | nmobndseqi 31315* | A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | nmobndseqiALT 31316* | Alternate shorter proof of nmobndseqi 31315 based on Axioms ax-reg 9564 and ax-ac2 10513 instead of ax-cc 10485. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | bloval 31317* | The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) | ||
| Theorem | isblo 31318 | The predicate "is a bounded linear operator." (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) | ||
| Theorem | isblo2 31319 | The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ))) | ||
| Theorem | bloln 31320 | A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿) | ||
| Theorem | blof 31321 | A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) | ||
| Theorem | nmblore 31322 | The norm of a bounded operator is a real number. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → (𝑁‘𝑇) ∈ ℝ) | ||
| Theorem | 0ofval 31323 | The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑊) & ⊢ 𝑂 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) | ||
| Theorem | 0oval 31324 | Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑊) & ⊢ 𝑂 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = 𝑍) | ||
| Theorem | 0oo 31325 | The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌) | ||
| Theorem | 0lno 31326 | The zero operator is linear. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐿) | ||
| Theorem | nmoo0 31327 | The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0) | ||
| Theorem | 0blo 31328 | The zero operator is a bounded linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 ∈ 𝐵) | ||
| Theorem | nmlno0lem 31329 | Lemma for nmlno0i 31330. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝑃 = (0vec‘𝑈) & ⊢ 𝑄 = (0vec‘𝑊) & ⊢ 𝐾 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) ⇒ ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍) | ||
| Theorem | nmlno0i 31330 | The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐿 → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlno0 31331 | The norm of a linear operator is zero iff the operator is zero. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)) | ||
| Theorem | nmlnoubi 31332* | An upper bound for the operator norm of a linear operator, using only the properties of nonzero arguments. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝐾 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ≠ 𝑍 → (𝑀‘(𝑇‘𝑥)) ≤ (𝐴 · (𝐾‘𝑥)))) → (𝑁‘𝑇) ≤ 𝐴) | ||
| Theorem | nmlnogt0 31333 | The norm of a nonzero linear operator is positive. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝑍 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇 ≠ 𝑍 ↔ 0 < (𝑁‘𝑇))) | ||
| Theorem | lnon0 31334* | The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑂 = (𝑈 0op 𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ 𝑇 ≠ 𝑂) → ∃𝑥 ∈ 𝑋 𝑥 ≠ 𝑍) | ||
| Theorem | nmblolbii 31335 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐵 ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | nmblolbi 31336 | A lower bound for the norm of a bounded linear operator. (Contributed by NM, 10-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐿 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝐴 ∈ 𝑋) → (𝑀‘(𝑇‘𝐴)) ≤ ((𝑁‘𝑇) · (𝐿‘𝐴))) | ||
| Theorem | isblo3i 31337* | The predicate "is a bounded linear operator." Definition 2.7-1 of [Kreyszig] p. 91. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (𝑀‘𝑦)))) | ||
| Theorem | blo3i 31338* | Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = (normCV‘𝑈) & ⊢ 𝑁 = (normCV‘𝑊) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐿 ∧ 𝐴 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑇‘𝑦)) ≤ (𝐴 · (𝑀‘𝑦))) → 𝑇 ∈ 𝐵) | ||
| Theorem | blometi 31339 | Upper bound for the distance between the values of a bounded linear operator. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ ((𝑇 ∈ 𝐵 ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → ((𝑇‘𝑃)𝐷(𝑇‘𝑄)) ≤ ((𝑁‘𝑇) · (𝑃𝐶𝑄))) | ||
| Theorem | blocnilem 31340 | Lemma for blocni 31341 and lnocni 31342. If a linear operator is continuous at any point, it is bounded. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵) | ||
| Theorem | blocni 31341 | A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 ⇒ ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) | ||
| Theorem | lnocni 31342 | If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐿 = (𝑈 LnOp 𝑊) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝑇 ∈ 𝐿 & ⊢ 𝑋 = (BaseSet‘𝑈) ⇒ ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | blocn 31343 | A linear operator is continuous iff it is bounded. Theorem 2.7-9(a) of [Kreyszig] p. 97. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec & ⊢ 𝐿 = (𝑈 LnOp 𝑊) ⇒ ⊢ (𝑇 ∈ 𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) | ||
| Theorem | blocn2 31344 | A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) & ⊢ 𝐵 = (𝑈 BLnOp 𝑊) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | ajfval 31345* | The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}) | ||
| Theorem | hmoval 31346* | The set of Hermitian (self-adjoint) operators on a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑡 ∈ dom 𝐴 ∣ (𝐴‘𝑡) = 𝑡}) | ||
| Theorem | ishmo 31347 | The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐻 = (HmOp‘𝑈) & ⊢ 𝐴 = (𝑈adj𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑇 ∈ 𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴‘𝑇) = 𝑇))) | ||
| Syntax | ccphlo 31348 | Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces). |
| class CPreHilOLD | ||
| Definition | df-ph 31349* | Define the class of all complex inner product spaces. An inner product space is a normed vector space whose norm satisfies the parallelogram law (a property that induces an inner product). Based on Exercise 4(b) of [ReedSimon] p. 63. The vector operation is 𝑔, the scalar product is 𝑠, and the norm is 𝑛. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) | ||
| Theorem | phnv 31350 | Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | ||
| Theorem | phrel 31351 | The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ Rel CPreHilOLD | ||
| Theorem | phnvi 31352 | Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ 𝑈 ∈ NrmCVec | ||
| Theorem | isphg 31353* | The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))) | ||
| Theorem | phop 31354 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩) | ||
| Theorem | cncph 31355 | The set of complex numbers is an inner product (pre-Hilbert) space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 7-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩ ⇒ ⊢ 𝑈 ∈ CPreHilOLD | ||
| Theorem | elimph 31356 | Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 | ||
| Theorem | elimphu 31357 | Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007.) (New usage is discouraged.) |
| ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHilOLD | ||
| Theorem | isph 31358* | The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))) | ||
| Theorem | phpar2 31359 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | phpar 31360 | The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
| Theorem | ip0i 31361 | A slight variant of Equation 6.46 of [Ponnusamy] p. 362, where 𝐽 is either 1 or -1 to represent +-1. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ ((((𝑁‘((𝐴𝐺𝐵)𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺𝐵)𝐺(-𝐽𝑆𝐶)))↑2)) + (((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘((𝐴𝐺(-1𝑆𝐵))𝐺(-𝐽𝑆𝐶)))↑2))) = (2 · (((𝑁‘(𝐴𝐺(𝐽𝑆𝐶)))↑2) − ((𝑁‘(𝐴𝐺(-𝐽𝑆𝐶)))↑2))) | ||
| Theorem | ip1ilem 31362 | Lemma for ip1i 31363. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐽 ∈ ℂ ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip1i 31363 | Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶)) | ||
| Theorem | ip2i 31364 | Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵)) | ||
| Theorem | ipdirilem 31365 | Lemma for ipdiri 31366. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)) | ||
| Theorem | ipdiri 31366 | Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
| Theorem | ipasslem1 31367 | Lemma for ipassi 31377. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem2 31368 | Lemma for ipassi 31377. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem3 31369 | Lemma for ipassi 31377. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem4 31370 | Lemma for ipassi 31377. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem5 31371 | Lemma for ipassi 31377. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐶 ∈ ℚ ∧ 𝐴 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem7 31372* | Lemma for ipassi 31377. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on ℝ. (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn 𝐾) | ||
| Theorem | ipasslem8 31373* | Lemma for ipassi 31377. By ipasslem5 31371, 𝐹 is 0 for all ℚ; since it is continuous and ℚ is dense in ℝ by qdensere2 25078, we conclude 𝐹 is 0 for all ℝ. (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) ⇒ ⊢ 𝐹:ℝ⟶{0} | ||
| Theorem | ipasslem9 31374 | Lemma for ipassi 31377. Conclude from ipasslem8 31373 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipasslem10 31375 | Lemma for ipassi 31377. Show the inner product associative law for the imaginary number i. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵)) | ||
| Theorem | ipasslem11 31376 | Lemma for ipassi 31377. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) | ||
| Theorem | ipassi 31377 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) | ||
| Theorem | dipdir 31378 | Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))) | ||
| Theorem | dipdi 31379 | Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶))) | ||
| Theorem | ip2dii 31380 | Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝐶 ∈ 𝑋 & ⊢ 𝐷 ∈ 𝑋 ⇒ ⊢ ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶))) | ||
| Theorem | dipass 31381 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) | ||
| Theorem | dipassr 31382 | "Associative" law for second argument of inner product (compare dipass 31381). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶))) | ||
| Theorem | dipassr2 31383 | "Associative" law for inner product. Conjugate version of dipassr 31382. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶))) | ||
| Theorem | dipsubdir 31384 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶))) | ||
| Theorem | dipsubdi 31385 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) | ||
| Theorem | pythi 31386 | The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space 𝑈. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) | ||
| Theorem | siilem1 31387 | Lemma for sii 31390. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐶 ∈ ℂ & ⊢ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ & ⊢ 0 ≤ (𝐶 · (𝐴𝑃𝐵)) ⇒ ⊢ ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) | ||
| Theorem | siilem2 31388 | Lemma for sii 31390. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))) | ||
| Theorem | siii 31389 | Inference from sii 31390. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐴 ∈ 𝑋 & ⊢ 𝐵 ∈ 𝑋 ⇒ ⊢ (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)) | ||
| Theorem | sii 31390 | Obsolete version of ipcau 25521 as of 22-Sep-2024. Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also Theorems bcseqi 31656, bcsiALT 31715, bcsiHIL 31716, csbren 25682. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) | ||
| Theorem | ipblnfi 31391* | A function 𝐹 generated by varying the first argument of an inner product (with its second argument a fixed vector 𝐴) is a bounded linear functional, i.e. a bounded linear operator from the vector space to ℂ. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝐶 = ⟨⟨ + , · ⟩, abs⟩ & ⊢ 𝐵 = (𝑈 BLnOp 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑥𝑃𝐴)) ⇒ ⊢ (𝐴 ∈ 𝑋 → 𝐹 ∈ 𝐵) | ||
| Theorem | ip2eqi 31392* | Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵)) | ||
| Theorem | phoeqi 31393* | A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇)) | ||
| Theorem | ajmoi 31394* | Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ CPreHilOLD ⇒ ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) | ||
| Theorem | ajfuni 31395 | The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑈adj𝑊) & ⊢ 𝑈 ∈ CPreHilOLD & ⊢ 𝑊 ∈ NrmCVec ⇒ ⊢ Fun 𝐴 | ||
| Theorem | ajfun 31396 | The adjoint function is a function. This is not immediately apparent from df-aj 31286 but results from the uniqueness shown by ajmoi 31394. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴) | ||
| Theorem | ajval 31397* | Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑌 = (BaseSet‘𝑊) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑄 = (·𝑖OLD‘𝑊) & ⊢ 𝐴 = (𝑈adj𝑊) ⇒ ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))) | ||
| Syntax | ccbn 31398 | Extend class notation with the class of all complex Banach spaces. |
| class CBan | ||
| Definition | df-cbn 31399 | Define the class of all complex Banach spaces. (Contributed by NM, 5-Dec-2006.) Use df-bn 25619 instead. (New usage is discouraged.) |
| ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈ (CMet‘(BaseSet‘𝑢))} | ||
| Theorem | iscbn 31400 | A complex Banach space is a normed complex vector space with a complete induced metric. (Contributed by NM, 5-Dec-2006.) Use isbn 25621 instead. (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ CBan ↔ (𝑈 ∈ NrmCVec ∧ 𝐷 ∈ (CMet‘𝑋))) | ||
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