Definition df-aj 28831

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 28829	. 2 class adj
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 28665	. . 3 class NrmCVec
5	2	cv 1542	. . . . . . 7 class 𝑢
6		cba 28667	. . . . . . 7 class BaseSet
7	5, 6	cfv 6380	. . . . . 6 class (BaseSet‘𝑢)
8	3	cv 1542	. . . . . . 7 class 𝑤
9	8, 6	cfv 6380	. . . . . 6 class (BaseSet‘𝑤)
10		vt	. . . . . . 7 setvar 𝑡
11	10	cv 1542	. . . . . 6 class 𝑡
12	7, 9, 11	wf 6376	. . . . 5 wff 𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤)
13		vs	. . . . . . 7 setvar 𝑠
14	13	cv 1542	. . . . . 6 class 𝑠
15	9, 7, 14	wf 6376	. . . . 5 wff 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢)
16		vx	. . . . . . . . . . 11 setvar 𝑥
17	16	cv 1542	. . . . . . . . . 10 class 𝑥
18	17, 11	cfv 6380	. . . . . . . . 9 class (𝑡‘𝑥)
19		vy	. . . . . . . . . 10 setvar 𝑦
20	19	cv 1542	. . . . . . . . 9 class 𝑦
21		cdip 28781	. . . . . . . . . 10 class ·𝑖OLD
22	8, 21	cfv 6380	. . . . . . . . 9 class (·𝑖OLD‘𝑤)
23	18, 20, 22	co 7213	. . . . . . . 8 class ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦)
24	20, 14	cfv 6380	. . . . . . . . 9 class (𝑠‘𝑦)
25	5, 21	cfv 6380	. . . . . . . . 9 class (·𝑖OLD‘𝑢)
26	17, 24, 25	co 7213	. . . . . . . 8 class (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
27	23, 26	wceq 1543	. . . . . . 7 wff ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
28	27, 19, 9	wral 3061	. . . . . 6 wff ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
29	28, 16, 7	wral 3061	. . . . 5 wff ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))
30	12, 15, 29	w3a 1089	. . . 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 5115	. . 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 7215	. 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 1543	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  28890