Definition df-aj 30836

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)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))})

Distinct variable group:   𝑡,𝑠,𝑢,𝑤,𝑥,𝑦

Detailed syntax breakdown of Definition df-aj

Step	Hyp	Ref	Expression
1		caj 30834	. 2 class adj
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30670	. . 3 class NrmCVec
5	2	cv 1541	. . . . . . 7 class 𝑢
6		cba 30672	. . . . . . 7 class BaseSet
7	5, 6	cfv 6492	. . . . . 6 class (BaseSet‘𝑢)
8	3	cv 1541	. . . . . . 7 class 𝑤
9	8, 6	cfv 6492	. . . . . 6 class (BaseSet‘𝑤)
10		vt	. . . . . . 7 setvar 𝑡
11	10	cv 1541	. . . . . 6 class 𝑡
12	7, 9, 11	wf 6488	. . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13		vs	. . . . . . 7 setvar 𝑠
14	13	cv 1541	. . . . . 6 class 𝑠
15	9, 7, 14	wf 6488	. . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16		vx	. . . . . . . . . . 11 setvar 𝑥
17	16	cv 1541	. . . . . . . . . 10 class 𝑥
18	17, 11	cfv 6492	. . . . . . . . 9 class (𝑡‘𝑥)
19		vy	. . . . . . . . . 10 setvar 𝑦
20	19	cv 1541	. . . . . . . . 9 class 𝑦
21		cdip 30786	. . . . . . . . . 10 class ·𝑖OLD
22	8, 21	cfv 6492	. . . . . . . . 9 class (·𝑖OLD‘𝑤)
23	18, 20, 22	co 7360	. . . . . . . 8 class ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦)
24	20, 14	cfv 6492	. . . . . . . . 9 class (𝑠‘𝑦)
25	5, 21	cfv 6492	. . . . . . . . 9 class (·𝑖OLD‘𝑢)
26	17, 24, 25	co 7360	. . . . . . . 8 class (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
27	23, 26	wceq 1542	. . . . . . 7 wff ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
28	27, 19, 9	wral 3052	. . . . . 6 wff ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
29	28, 16, 7	wral 3052	. . . . 5 wff ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
30	12, 15, 29	w3a 1087	. . . 4 wff (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))
31	30, 10, 13	copab 5148	. . 3 class {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))}
32	2, 3, 4, 4, 31	cmpo 7362	. 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))})
33	1, 32	wceq 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)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))})

This definition is referenced by:  ajfval  30895