Definition df-aj 31080

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 31078	. 2 class adj
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30914	. . 3 class NrmCVec
5	2	cv 1569	. . . . . . 7 class 𝑢
6		cba 30916	. . . . . . 7 class BaseSet
7	5, 6	cfv 6538	. . . . . 6 class (BaseSet‘𝑢)
8	3	cv 1569	. . . . . . 7 class 𝑤
9	8, 6	cfv 6538	. . . . . 6 class (BaseSet‘𝑤)
10		vt	. . . . . . 7 setvar 𝑡
11	10	cv 1569	. . . . . 6 class 𝑡
12	7, 9, 11	wf 6534	. . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13		vs	. . . . . . 7 setvar 𝑠
14	13	cv 1569	. . . . . 6 class 𝑠
15	9, 7, 14	wf 6534	. . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16		vx	. . . . . . . . . . 11 setvar 𝑥
17	16	cv 1569	. . . . . . . . . 10 class 𝑥
18	17, 11	cfv 6538	. . . . . . . . 9 class (𝑡‘𝑥)
19		vy	. . . . . . . . . 10 setvar 𝑦
20	19	cv 1569	. . . . . . . . 9 class 𝑦
21		cdip 31030	. . . . . . . . . 10 class ·𝑖OLD
22	8, 21	cfv 6538	. . . . . . . . 9 class (·𝑖OLD‘𝑤)
23	18, 20, 22	co 7412	. . . . . . . 8 class ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦)
24	20, 14	cfv 6538	. . . . . . . . 9 class (𝑠‘𝑦)
25	5, 21	cfv 6538	. . . . . . . . 9 class (·𝑖OLD‘𝑢)
26	17, 24, 25	co 7412	. . . . . . . 8 class (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
27	23, 26	wceq 1570	. . . . . . 7 wff ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
28	27, 19, 9	wral 3079	. . . . . 6 wff ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
29	28, 16, 7	wral 3079	. . . . 5 wff ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
30	12, 15, 29	w3a 1103	. . . 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 5174	. . 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 7414	. 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 1570	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  31139