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Definition df-aj 26989
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 26987 . 2 class adj
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 26823 . . 3 class NrmCVec
52cv 1474 . . . . . . 7 class 𝑢
6 cba 26825 . . . . . . 7 class BaseSet
75, 6cfv 5804 . . . . . 6 class (BaseSet‘𝑢)
83cv 1474 . . . . . . 7 class 𝑤
98, 6cfv 5804 . . . . . 6 class (BaseSet‘𝑤)
10 vt . . . . . . 7 setvar 𝑡
1110cv 1474 . . . . . 6 class 𝑡
127, 9, 11wf 5800 . . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13 vs . . . . . . 7 setvar 𝑠
1413cv 1474 . . . . . 6 class 𝑠
159, 7, 14wf 5800 . . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16 vx . . . . . . . . . . 11 setvar 𝑥
1716cv 1474 . . . . . . . . . 10 class 𝑥
1817, 11cfv 5804 . . . . . . . . 9 class (𝑡𝑥)
19 vy . . . . . . . . . 10 setvar 𝑦
2019cv 1474 . . . . . . . . 9 class 𝑦
21 cdip 26939 . . . . . . . . . 10 class ·𝑖OLD
228, 21cfv 5804 . . . . . . . . 9 class (·𝑖OLD𝑤)
2318, 20, 22co 6549 . . . . . . . 8 class ((𝑡𝑥)(·𝑖OLD𝑤)𝑦)
2420, 14cfv 5804 . . . . . . . . 9 class (𝑠𝑦)
255, 21cfv 5804 . . . . . . . . 9 class (·𝑖OLD𝑢)
2617, 24, 25co 6549 . . . . . . . 8 class (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2723, 26wceq 1475 . . . . . . 7 wff ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2827, 19, 9wral 2896 . . . . . 6 wff 𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
2928, 16, 7wral 2896 . . . . 5 wff 𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))
3012, 15, 29w3a 1031 . . . 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 4642 . . 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 6551 . 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 1475 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  27048
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