Definition df-nmoo 30725

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 30721	. 2 class normOpOLD
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30564	. . 3 class NrmCVec
5		vt	. . . 4 setvar 𝑡
6	3	cv 1540	. . . . . 6 class 𝑤
7		cba 30566	. . . . . 6 class BaseSet
8	6, 7	cfv 6481	. . . . 5 class (BaseSet‘𝑤)
9	2	cv 1540	. . . . . 6 class 𝑢
10	9, 7	cfv 6481	. . . . 5 class (BaseSet‘𝑢)
11		cmap 8750	. . . . 5 class ↑m
12	8, 10, 11	co 7346	. . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13		vz	. . . . . . . . . . 11 setvar 𝑧
14	13	cv 1540	. . . . . . . . . 10 class 𝑧
15		cnmcv 30570	. . . . . . . . . . 11 class normCV
16	9, 15	cfv 6481	. . . . . . . . . 10 class (normCV‘𝑢)
17	14, 16	cfv 6481	. . . . . . . . 9 class ((normCV‘𝑢)‘𝑧)
18		c1 11007	. . . . . . . . 9 class 1
19		cle 11147	. . . . . . . . 9 class ≤
20	17, 18, 19	wbr 5089	. . . . . . . 8 wff ((normCV‘𝑢)‘𝑧) ≤ 1
21		vx	. . . . . . . . . 10 setvar 𝑥
22	21	cv 1540	. . . . . . . . 9 class 𝑥
23	5	cv 1540	. . . . . . . . . . 11 class 𝑡
24	14, 23	cfv 6481	. . . . . . . . . 10 class (𝑡‘𝑧)
25	6, 15	cfv 6481	. . . . . . . . . 10 class (normCV‘𝑤)
26	24, 25	cfv 6481	. . . . . . . . 9 class ((normCV‘𝑤)‘(𝑡‘𝑧))
27	22, 26	wceq 1541	. . . . . . . 8 wff 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))
28	20, 27	wa 395	. . . . . . 7 wff (((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
29	28, 13, 10	wrex 3056	. . . . . 6 wff ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
30	29, 21	cab 2709	. . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}
31		cxr 11145	. . . . 5 class ℝ*
32		clt 11146	. . . . 5 class <
33	30, 31, 32	csup 9324	. . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )
34	5, 12, 33	cmpt 5170	. . 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 7348	. 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 1541	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 referenced by:  nmoofval  30742