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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmoofval 27001* The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))

Theoremnmooval 27002* The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))

Theoremnmosetre 27003* The set in the supremum of the operator norm definition df-nmoo 26984 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
𝑌 = (BaseSet‘𝑊)    &   𝑁 = (normCV𝑊)       ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))} ⊆ ℝ)

Theoremnmosetn0 27004* The set in the supremum of the operator norm definition df-nmoo 26984 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑀 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (𝑁‘(𝑇𝑍)) ∈ {𝑥 ∣ ∃𝑦𝑋 ((𝑀𝑦) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑦)))})

Theoremnmoxr 27005 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:𝑋𝑌) → (𝑁𝑇) ∈ ℝ*)

Theoremnmooge0 27006 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:𝑋𝑌) → 0 ≤ (𝑁𝑇))

Theoremnmorepnf 27007 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:𝑋𝑌) → ((𝑁𝑇) ∈ ℝ ↔ (𝑁𝑇) ≠ +∞))

Theoremnmoreltpnf 27008 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:𝑋𝑌) → ((𝑁𝑇) ∈ ℝ ↔ (𝑁𝑇) < +∞))

Theoremnmogtmnf 27009 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:𝑋𝑌) → -∞ < (𝑁𝑇))

Theoremnmoolb 27010 A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑀‘(𝑇𝐴)) ≤ (𝑁𝑇))

Theoremnmoubi 27011* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))

Theoremnmoub3i 27012* An upper bound for an operator norm. (Contributed by NM, 12-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌𝐴 ∈ ℝ ∧ ∀𝑥𝑋 (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐿𝑥))) → (𝑁𝑇) ≤ (abs‘𝐴))

Theoremnmoub2i 27013* An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥𝑋 (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐿𝑥))) → (𝑁𝑇) ≤ 𝐴)

Theoremnmobndi 27014* 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑟)))

Theoremnmounbi 27015* 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇:𝑋𝑌 → ((𝑁𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇𝑦)))))

Theoremnmounbseqi 27016* 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝑁𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓𝑘))))))

TheoremnmounbseqiALT 27017* Alternate shorter proof of nmounbseqi 27016 based on axioms ax-reg 8380 and ax-ac2 9168 instead of ax-cc 9140. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ (𝑁𝑇) = +∞) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 ∧ 𝑘 < (𝑀‘(𝑇‘(𝑓𝑘))))))

Theoremnmobndseqi 27018* 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)

TheoremnmobndseqiALT 27019* Alternate shorter proof of nmobndseqi 27018 based on axioms ax-reg 8380 and ax-ac2 9168 instead of ax-cc 9140. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝐿 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)

Theorembloval 27020* 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 = {𝑡𝐿 ∣ (𝑁𝑡) < +∞})

Theoremisblo 27021 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) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) < +∞)))

Theoremisblo2 27022 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) → (𝑇𝐵 ↔ (𝑇𝐿 ∧ (𝑁𝑇) ∈ ℝ)))

Theorembloln 27023 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 ∧ 𝑇𝐵) → 𝑇𝐿)

Theoremblof 27024 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 ∧ 𝑇𝐵) → 𝑇:𝑋𝑌)

Theoremnmblore 27025 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 ∧ 𝑇𝐵) → (𝑁𝑇) ∈ ℝ)

Theorem0ofval 27026 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 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Theorem0oval 27027 Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑊)    &   𝑂 = (𝑈 0op 𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑂𝐴) = 𝑍)

Theorem0oo 27028 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:𝑋𝑌)

Theorem0lno 27029 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) → 𝑍𝐿)

Theoremnmoo0 27030 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) → (𝑁𝑍) = 0)

Theorem0blo 27031 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) → 𝑍𝐵)

Theoremnmlno0lem 27032 Lemma for nmlno0i 27033. (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    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)    &   𝑌 = (BaseSet‘𝑊)    &   𝑅 = ( ·𝑠OLD𝑈)    &   𝑆 = ( ·𝑠OLD𝑊)    &   𝑃 = (0vec𝑈)    &   𝑄 = (0vec𝑊)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)       ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍)

Theoremnmlno0i 27033 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       (𝑇𝐿 → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlno0 27034 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 ∧ 𝑇𝐿) → ((𝑁𝑇) = 0 ↔ 𝑇 = 𝑍))

Theoremnmlnoubi 27035* 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 = (0vec𝑈)    &   𝐾 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿 ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥𝑋 (𝑥𝑍 → (𝑀‘(𝑇𝑥)) ≤ (𝐴 · (𝐾𝑥)))) → (𝑁𝑇) ≤ 𝐴)

Theoremnmlnogt0 27036 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 ∧ 𝑇𝐿) → (𝑇𝑍 ↔ 0 < (𝑁𝑇)))

Theoremlnon0 27037* 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 = (0vec𝑈)    &   𝑂 = (𝑈 0op 𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)       (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ 𝑇𝑂) → ∃𝑥𝑋 𝑥𝑍)

Theoremnmblolbii 27038 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec    &   𝑇𝐵       (𝐴𝑋 → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremnmblolbi 27039 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 = (normCV𝑈)    &   𝑀 = (normCV𝑊)    &   𝑁 = (𝑈 normOpOLD 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐵𝐴𝑋) → (𝑀‘(𝑇𝐴)) ≤ ((𝑁𝑇) · (𝐿𝐴)))

Theoremisblo3i 27040* 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 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       (𝑇𝐵 ↔ (𝑇𝐿 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (𝑀𝑦))))

Theoremblo3i 27041* Properties that determine a bounded linear operator. (Contributed by NM, 13-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = (normCV𝑈)    &   𝑁 = (normCV𝑊)    &   𝐿 = (𝑈 LnOp 𝑊)    &   𝐵 = (𝑈 BLnOp 𝑊)    &   𝑈 ∈ NrmCVec    &   𝑊 ∈ NrmCVec       ((𝑇𝐿𝐴 ∈ ℝ ∧ ∀𝑦𝑋 (𝑁‘(𝑇𝑦)) ≤ (𝐴 · (𝑀𝑦))) → 𝑇𝐵)

Theoremblometi 27042 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       ((𝑇𝐵𝑃𝑋𝑄𝑋) → ((𝑇𝑃)𝐷(𝑇𝑄)) ≤ ((𝑁𝑇) · (𝑃𝐶𝑄)))

Theoremblocnilem 27043 Lemma for blocni 27044 and lnocni 27045. 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    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)       ((𝑃𝑋𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇𝐵)

Theoremblocni 27044 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    &   𝑇𝐿       (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇𝐵)

Theoremlnocni 27045 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    &   𝑇𝐿    &   𝑋 = (BaseSet‘𝑈)       ((𝑃𝑋𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾))

Theoremblocn 27046 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)       (𝑇𝐿 → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇𝐵))

Theoremblocn2 27047 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       (𝑇𝐵𝑇 ∈ (𝐽 Cn 𝐾))

Theoremajfval 27048* 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 = (·𝑖OLD𝑈)    &   𝑄 = (·𝑖OLD𝑊)    &   𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})

Theoremhmoval 27049* 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 ∣ (𝐴𝑡) = 𝑡})

Theoremishmo 27050 The predicate "is a hermitian operator." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐻 = (HmOp‘𝑈)    &   𝐴 = (𝑈adj𝑈)       (𝑈 ∈ NrmCVec → (𝑇𝐻 ↔ (𝑇 ∈ dom 𝐴 ∧ (𝐴𝑇) = 𝑇)))

19.5  Inner product (pre-Hilbert) spaces

19.5.1  Definition and basic properties

Syntaxccphlo 27051 Extend class notation with the class of all complex inner product spaces (also called pre-Hilbert spaces).
class CPreHilOLD

Definitiondf-ph 27052* 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 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})

Theoremphnv 27053 Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
(𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)

Theoremphrel 27054 The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Rel CPreHilOLD

Theoremphnvi 27055 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

Theoremisphg 27056* 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       ((𝐺𝐴𝑆𝐵𝑁𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))

Theoremphop 27057 A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

19.5.2  Examples of pre-Hilbert spaces

Theoremcncph 27058 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

Theoremelimph 27059 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 = (0vec𝑈)    &   𝑈 ∈ CPreHilOLD       if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Theoremelimphu 27060 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

19.5.3  Properties of pre-Hilbert spaces

Theoremisph 27061* The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)))))

Theoremphpar2 27062 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremphpar 27063 The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremip0i 27064 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   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)))

Theoremip1ilem 27065 Lemma for ip1i 27066. (Contributed by NM, 21-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝑁 = (normCV𝑈)    &   𝐽 ∈ ℂ       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip1i 27066 Equation 6.47 of [Ponnusamy] p. 362. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       (((𝐴𝐺𝐵)𝑃𝐶) + ((𝐴𝐺(-1𝑆𝐵))𝑃𝐶)) = (2 · (𝐴𝑃𝐶))

Theoremip2i 27067 Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((2𝑆𝐴)𝑃𝐵) = (2 · (𝐴𝑃𝐵))

Theoremipdirilem 27068 Lemma for ipdiri 27069. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋       ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶))

Theoremipdiri 27069 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋𝐶𝑋) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremipasslem1 27070 Lemma for ipassi 27080. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem2 27071 Lemma for ipassi 27080. Show the inner product associative law for nonpositive integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ0𝐴𝑋) → ((-𝑁𝑆𝐴)𝑃𝐵) = (-𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem3 27072 Lemma for ipassi 27080. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℤ ∧ 𝐴𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Theoremipasslem4 27073 Lemma for ipassi 27080. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝑁 ∈ ℕ ∧ 𝐴𝑋) → (((1 / 𝑁)𝑆𝐴)𝑃𝐵) = ((1 / 𝑁) · (𝐴𝑃𝐵)))

Theoremipasslem5 27074 Lemma for ipassi 27080. Show the inner product associative law for rational numbers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐵𝑋       ((𝐶 ∈ ℚ ∧ 𝐴𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem7 27075* Lemma for ipassi 27080. Show that ((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)) is continuous on . (Contributed by NM, 23-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ (𝐽 Cn 𝐾)

Theoremipasslem8 27076* Lemma for ipassi 27080. By ipasslem5 27074, 𝐹 is 0 for all ; since it is continuous and is dense in by qdensere2 22408, we conclude 𝐹 is 0 for all . (Contributed by NM, 24-Aug-2007.) (Revised by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐹 = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))       𝐹:ℝ⟶{0}

Theoremipasslem9 27077 Lemma for ipassi 27080. Conclude from ipasslem8 27076 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipasslem10 27078 Lemma for ipassi 27080. 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑁 = (normCV𝑈)       ((i𝑆𝐴)𝑃𝐵) = (i · (𝐴𝑃𝐵))

Theoremipasslem11 27079 Lemma for ipassi 27080. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))

Theoremipassi 27080 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 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipdir 27081 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 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) + (𝐵𝑃𝐶)))

Theoremdipdi 27082 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝐺𝐶)) = ((𝐴𝑃𝐵) + (𝐴𝑃𝐶)))

Theoremip2dii 27083 Inner product of two sums. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝐶𝑋    &   𝐷𝑋       ((𝐴𝐺𝐵)𝑃(𝐶𝐺𝐷)) = (((𝐴𝑃𝐶) + (𝐵𝑃𝐷)) + ((𝐴𝑃𝐷) + (𝐵𝑃𝐶)))

Theoremdipass 27084 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 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Theoremdipassr 27085 "Associative" law for second argument of inner product (compare dipass 27084). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃(𝐵𝑆𝐶)) = ((∗‘𝐵) · (𝐴𝑃𝐶)))

Theoremdipassr2 27086 "Associative" law for inner product. Conjugate version of dipassr 27085. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑃((∗‘𝐵)𝑆𝐶)) = (𝐵 · (𝐴𝑃𝐶)))

Theoremdipsubdir 27087 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑃𝐶) = ((𝐴𝑃𝐶) − (𝐵𝑃𝐶)))

Theoremdipsubdi 27088 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑃 = (·𝑖OLD𝑈)       ((𝑈 ∈ CPreHilOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶)))

Theorempythi 27089 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. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))

Theoremsiilem1 27090 Lemma for sii 27093. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝐶 ∈ ℂ    &   (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ    &   0 ≤ (𝐶 · (𝐴𝑃𝐵))       ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵)))

Theoremsiilem2 27091 Lemma for sii 27093. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁𝐵)↑2)))) ≤ ((𝑁𝐴) · (𝑁𝐵))))

Theoremsiii 27092 Inference from sii 27093. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐴𝑋    &   𝐵𝑋       (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵))

Theoremsii 27093 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 27361, bcsiALT 27420, bcsiHIL 27421, csbren 22990. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁𝐴) · (𝑁𝐵)))

Theoremsspph 27094 A subspace of an inner product space is an inner product space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
𝐻 = (SubSp‘𝑈)       ((𝑈 ∈ CPreHilOLD𝑊𝐻) → 𝑊 ∈ CPreHilOLD)

Theoremipblnfi 27095* 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 . (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD    &   𝐶 = ⟨⟨ + , · ⟩, abs⟩    &   𝐵 = (𝑈 BLnOp 𝐶)    &   𝐹 = (𝑥𝑋 ↦ (𝑥𝑃𝐴))       (𝐴𝑋𝐹𝐵)

Theoremip2eqi 27096* 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 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵))

Theoremphoeqi 27097* A condition implying that two operators are equal. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ((𝑆:𝑌𝑋𝑇:𝑌𝑋) → (∀𝑥𝑋𝑦𝑌 (𝑥𝑃(𝑆𝑦)) = (𝑥𝑃(𝑇𝑦)) ↔ 𝑆 = 𝑇))

Theoremajmoi 27098* Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑃 = (·𝑖OLD𝑈)    &   𝑈 ∈ CPreHilOLD       ∃*𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))

Theoremajfuni 27099 The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)    &   𝑈 ∈ CPreHilOLD    &   𝑊 ∈ NrmCVec       Fun 𝐴

Theoremajfun 27100 The adjoint function is a function. This is not immediately apparent from df-aj 26989 but results from the uniqueness shown by ajmoi 27098. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
𝐴 = (𝑈adj𝑊)       ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → Fun 𝐴)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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