Definition df-nmoo 31106

Description: Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces ⟨𝑢, 𝑤⟩. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
df-nmoo	⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))

Distinct variable group:   𝑢,𝑡,𝑤,𝑥,𝑧

Detailed syntax breakdown of Definition df-nmoo

Step	Hyp	Ref	Expression
1		cnmoo 31102	. 2 class normOpOLD
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30945	. . 3 class NrmCVec
5		vt	. . . 4 setvar 𝑡
6	3	cv 1569	. . . . . 6 class 𝑤
7		cba 30947	. . . . . 6 class BaseSet
8	6, 7	cfv 6536	. . . . 5 class (BaseSet‘𝑤)
9	2	cv 1569	. . . . . 6 class 𝑢
10	9, 7	cfv 6536	. . . . 5 class (BaseSet‘𝑢)
11		cmap 8820	. . . . 5 class ↑m
12	8, 10, 11	co 7410	. . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13		vz	. . . . . . . . . . 11 setvar 𝑧
14	13	cv 1569	. . . . . . . . . 10 class 𝑧
15		cnmcv 30951	. . . . . . . . . . 11 class normCV
16	9, 15	cfv 6536	. . . . . . . . . 10 class (normCV‘𝑢)
17	14, 16	cfv 6536	. . . . . . . . 9 class ((normCV‘𝑢)‘𝑧)
18		c1 11105	. . . . . . . . 9 class 1
19		cle 11248	. . . . . . . . 9 class ≤
20	17, 18, 19	wbr 5109	. . . . . . . 8 wff ((normCV‘𝑢)‘𝑧) ≤ 1
21		vx	. . . . . . . . . 10 setvar 𝑥
22	21	cv 1569	. . . . . . . . 9 class 𝑥
23	5	cv 1569	. . . . . . . . . . 11 class 𝑡
24	14, 23	cfv 6536	. . . . . . . . . 10 class (𝑡‘𝑧)
25	6, 15	cfv 6536	. . . . . . . . . 10 class (normCV‘𝑤)
26	24, 25	cfv 6536	. . . . . . . . 9 class ((normCV‘𝑤)‘(𝑡‘𝑧))
27	22, 26	wceq 1570	. . . . . . . 8 wff 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))
28	20, 27	wa 400	. . . . . . 7 wff (((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
29	28, 13, 10	wrex 3089	. . . . . 6 wff ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
30	29, 21	cab 2741	. . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}
31		cxr 11246	. . . . 5 class ℝ*
32		clt 11247	. . . . 5 class <
33	30, 31, 32	csup 9396	. . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )
34	5, 12, 33	cmpt 5192	. . 3 class (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ))
35	2, 3, 4, 4, 34	cmpo 7412	. 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
36	1, 35	wceq 1570	1 wff normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))

This definition is used by:  nmoofval  31123