Definition df-nmo 24729

Description: Define the norm of an operator between two normed groups (usually vector spaces). This definition produces an operator norm function for each pair of groups ⟨𝑠, 𝑡⟩. Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 25-Sep-2020.)

Assertion

Ref	Expression
df-nmo	⊢ normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))

Distinct variable group:   𝑓,𝑟,𝑠,𝑡,𝑥

Detailed syntax breakdown of Definition df-nmo

Step	Hyp	Ref	Expression
1		cnmo 24726	. 2 class normOp
2		vs	. . 3 setvar 𝑠
3		vt	. . 3 setvar 𝑡
4		cngp 24590	. . 3 class NrmGrp
5		vf	. . . 4 setvar 𝑓
6	2	cv 1539	. . . . 5 class 𝑠
7	3	cv 1539	. . . . 5 class 𝑡
8		cghm 19230	. . . . 5 class GrpHom
9	6, 7, 8	co 7431	. . . 4 class (𝑠 GrpHom 𝑡)
10		vx	. . . . . . . . . . 11 setvar 𝑥
11	10	cv 1539	. . . . . . . . . 10 class 𝑥
12	5	cv 1539	. . . . . . . . . 10 class 𝑓
13	11, 12	cfv 6561	. . . . . . . . 9 class (𝑓‘𝑥)
14		cnm 24589	. . . . . . . . . 10 class norm
15	7, 14	cfv 6561	. . . . . . . . 9 class (norm‘𝑡)
16	13, 15	cfv 6561	. . . . . . . 8 class ((norm‘𝑡)‘(𝑓‘𝑥))
17		vr	. . . . . . . . . 10 setvar 𝑟
18	17	cv 1539	. . . . . . . . 9 class 𝑟
19	6, 14	cfv 6561	. . . . . . . . . 10 class (norm‘𝑠)
20	11, 19	cfv 6561	. . . . . . . . 9 class ((norm‘𝑠)‘𝑥)
21		cmul 11160	. . . . . . . . 9 class ·
22	18, 20, 21	co 7431	. . . . . . . 8 class (𝑟 · ((norm‘𝑠)‘𝑥))
23		cle 11296	. . . . . . . 8 class ≤
24	16, 22, 23	wbr 5143	. . . . . . 7 wff ((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))
25		cbs 17247	. . . . . . . 8 class Base
26	6, 25	cfv 6561	. . . . . . 7 class (Base‘𝑠)
27	24, 10, 26	wral 3061	. . . . . 6 wff ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))
28		cc0 11155	. . . . . . 7 class 0
29		cpnf 11292	. . . . . . 7 class +∞
30		cico 13389	. . . . . . 7 class [,)
31	28, 29, 30	co 7431	. . . . . 6 class (0[,)+∞)
32	27, 17, 31	crab 3436	. . . . 5 class {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}
33		cxr 11294	. . . . 5 class ℝ*
34		clt 11295	. . . . 5 class <
35	32, 33, 34	cinf 9481	. . . 4 class inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )
36	5, 9, 35	cmpt 5225	. . 3 class (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < ))
37	2, 3, 4, 4, 36	cmpo 7433	. 2 class (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
38	1, 37	wceq 1540	1 wff normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))

This definition is referenced by:  nmoffn  24732  nmofval  24735