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Definition df-aj 27885
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 27883 . 2 class adj
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 27719 . . 3 class NrmCVec
52cv 1619 . . . . . . 7 class 𝑢
6 cba 27721 . . . . . . 7 class BaseSet
75, 6cfv 6037 . . . . . 6 class (BaseSet‘𝑢)
83cv 1619 . . . . . . 7 class 𝑤
98, 6cfv 6037 . . . . . 6 class (BaseSet‘𝑤)
10 vt . . . . . . 7 setvar 𝑡
1110cv 1619 . . . . . 6 class 𝑡
127, 9, 11wf 6033 . . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13 vs . . . . . . 7 setvar 𝑠
1413cv 1619 . . . . . 6 class 𝑠
159, 7, 14wf 6033 . . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16 vx . . . . . . . . . . 11 setvar 𝑥
1716cv 1619 . . . . . . . . . 10 class 𝑥
1817, 11cfv 6037 . . . . . . . . 9 class (𝑡𝑥)
19 vy . . . . . . . . . 10 setvar 𝑦
2019cv 1619 . . . . . . . . 9 class 𝑦
21 cdip 27835 . . . . . . . . . 10 class ·𝑖OLD
228, 21cfv 6037 . . . . . . . . 9 class (·𝑖OLD𝑤)
2318, 20, 22co 6801 . . . . . . . 8 class ((𝑡𝑥)(·𝑖OLD𝑤)𝑦)
2420, 14cfv 6037 . . . . . . . . 9 class (𝑠𝑦)
255, 21cfv 6037 . . . . . . . . 9 class (·𝑖OLD𝑢)
2617, 24, 25co 6801 . . . . . . . 8 class (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2723, 26wceq 1620 . . . . . . 7 wff ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2827, 19, 9wral 3038 . . . . . 6 wff 𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2928, 16, 7wral 3038 . . . . 5 wff 𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
3012, 15, 29w3a 1072 . . . 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 4852 . . 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, 31cmpt2 6803 . 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 1620 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  27944
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