Definition df-nmoo 30835

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 30831	. 2 class normOpOLD
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30674	. . 3 class NrmCVec
5		vt	. . . 4 setvar 𝑡
6	3	cv 1546	. . . . . 6 class 𝑤
7		cba 30676	. . . . . 6 class BaseSet
8	6, 7	cfv 6486	. . . . 5 class (BaseSet‘𝑤)
9	2	cv 1546	. . . . . 6 class 𝑢
10	9, 7	cfv 6486	. . . . 5 class (BaseSet‘𝑢)
11		cmap 8764	. . . . 5 class ↑m
12	8, 10, 11	co 7357	. . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13		vz	. . . . . . . . . . 11 setvar 𝑧
14	13	cv 1546	. . . . . . . . . 10 class 𝑧
15		cnmcv 30680	. . . . . . . . . . 11 class normCV
16	9, 15	cfv 6486	. . . . . . . . . 10 class (normCV‘𝑢)
17	14, 16	cfv 6486	. . . . . . . . 9 class ((normCV‘𝑢)‘𝑧)
18		c1 11031	. . . . . . . . 9 class 1
19		cle 11172	. . . . . . . . 9 class ≤
20	17, 18, 19	wbr 5073	. . . . . . . 8 wff ((normCV‘𝑢)‘𝑧) ≤ 1
21		vx	. . . . . . . . . 10 setvar 𝑥
22	21	cv 1546	. . . . . . . . 9 class 𝑥
23	5	cv 1546	. . . . . . . . . . 11 class 𝑡
24	14, 23	cfv 6486	. . . . . . . . . 10 class (𝑡‘𝑧)
25	6, 15	cfv 6486	. . . . . . . . . 10 class (normCV‘𝑤)
26	24, 25	cfv 6486	. . . . . . . . 9 class ((normCV‘𝑤)‘(𝑡‘𝑧))
27	22, 26	wceq 1547	. . . . . . . 8 wff 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))
28	20, 27	wa 396	. . . . . . 7 wff (((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
29	28, 13, 10	wrex 3063	. . . . . 6 wff ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
30	29, 21	cab 2717	. . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}
31		cxr 11170	. . . . 5 class ℝ*
32		clt 11171	. . . . 5 class <
33	30, 31, 32	csup 9344	. . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )
34	5, 12, 33	cmpt 5154	. . 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 7359	. 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 1547	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  30852