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Definition df-aj 30559
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 30557 . 2 class adj
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 30393 . . 3 class NrmCVec
52cv 1533 . . . . . . 7 class 𝑢
6 cba 30395 . . . . . . 7 class BaseSet
75, 6cfv 6548 . . . . . 6 class (BaseSet‘𝑢)
83cv 1533 . . . . . . 7 class 𝑤
98, 6cfv 6548 . . . . . 6 class (BaseSet‘𝑤)
10 vt . . . . . . 7 setvar 𝑡
1110cv 1533 . . . . . 6 class 𝑡
127, 9, 11wf 6544 . . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13 vs . . . . . . 7 setvar 𝑠
1413cv 1533 . . . . . 6 class 𝑠
159, 7, 14wf 6544 . . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16 vx . . . . . . . . . . 11 setvar 𝑥
1716cv 1533 . . . . . . . . . 10 class 𝑥
1817, 11cfv 6548 . . . . . . . . 9 class (𝑡𝑥)
19 vy . . . . . . . . . 10 setvar 𝑦
2019cv 1533 . . . . . . . . 9 class 𝑦
21 cdip 30509 . . . . . . . . . 10 class ·𝑖OLD
228, 21cfv 6548 . . . . . . . . 9 class (·𝑖OLD𝑤)
2318, 20, 22co 7420 . . . . . . . 8 class ((𝑡𝑥)(·𝑖OLD𝑤)𝑦)
2420, 14cfv 6548 . . . . . . . . 9 class (𝑠𝑦)
255, 21cfv 6548 . . . . . . . . 9 class (·𝑖OLD𝑢)
2617, 24, 25co 7420 . . . . . . . 8 class (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2723, 26wceq 1534 . . . . . . 7 wff ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2827, 19, 9wral 3058 . . . . . 6 wff 𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2928, 16, 7wral 3058 . . . . 5 wff 𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
3012, 15, 29w3a 1085 . . . 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 5210 . . 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 7422 . 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 1534 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  30618
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