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Definition df-aj 30003
Description: Define the adjoint of an operator (if it exists). The domain of 𝑈adj𝑊 is the set of all operators from 𝑈 to 𝑊 that have an adjoint. Definition 3.9-1 of [Kreyszig] p. 196, although we don't require that 𝑈 and 𝑊 be Hilbert spaces nor that the operators be linear. Although we define it for any normed vector space for convenience, the definition is meaningful only for inner product spaces. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-aj adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})
Distinct variable group:   𝑡,𝑠,𝑢,𝑤,𝑥,𝑦

Detailed syntax breakdown of Definition df-aj
StepHypRef Expression
1 caj 30001 . 2 class adj
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 29837 . . 3 class NrmCVec
52cv 1541 . . . . . . 7 class 𝑢
6 cba 29839 . . . . . . 7 class BaseSet
75, 6cfv 6544 . . . . . 6 class (BaseSet‘𝑢)
83cv 1541 . . . . . . 7 class 𝑤
98, 6cfv 6544 . . . . . 6 class (BaseSet‘𝑤)
10 vt . . . . . . 7 setvar 𝑡
1110cv 1541 . . . . . 6 class 𝑡
127, 9, 11wf 6540 . . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13 vs . . . . . . 7 setvar 𝑠
1413cv 1541 . . . . . 6 class 𝑠
159, 7, 14wf 6540 . . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16 vx . . . . . . . . . . 11 setvar 𝑥
1716cv 1541 . . . . . . . . . 10 class 𝑥
1817, 11cfv 6544 . . . . . . . . 9 class (𝑡𝑥)
19 vy . . . . . . . . . 10 setvar 𝑦
2019cv 1541 . . . . . . . . 9 class 𝑦
21 cdip 29953 . . . . . . . . . 10 class ·𝑖OLD
228, 21cfv 6544 . . . . . . . . 9 class (·𝑖OLD𝑤)
2318, 20, 22co 7409 . . . . . . . 8 class ((𝑡𝑥)(·𝑖OLD𝑤)𝑦)
2420, 14cfv 6544 . . . . . . . . 9 class (𝑠𝑦)
255, 21cfv 6544 . . . . . . . . 9 class (·𝑖OLD𝑢)
2617, 24, 25co 7409 . . . . . . . 8 class (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2723, 26wceq 1542 . . . . . . 7 wff ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2827, 19, 9wral 3062 . . . . . 6 wff 𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2928, 16, 7wral 3062 . . . . 5 wff 𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
3012, 15, 29w3a 1088 . . . 4 wff (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))
3130, 10, 13copab 5211 . . 3 class {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))}
322, 3, 4, 4, 31cmpo 7411 . 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})
331, 32wceq 1542 1 wff adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})
Colors of variables: wff setvar class
This definition is referenced by:  ajfval  30062
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