Definition df-nmoo 30681

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 30677	. 2 class normOpOLD
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30520	. . 3 class NrmCVec
5		vt	. . . 4 setvar 𝑡
6	3	cv 1539	. . . . . 6 class 𝑤
7		cba 30522	. . . . . 6 class BaseSet
8	6, 7	cfv 6514	. . . . 5 class (BaseSet‘𝑤)
9	2	cv 1539	. . . . . 6 class 𝑢
10	9, 7	cfv 6514	. . . . 5 class (BaseSet‘𝑢)
11		cmap 8802	. . . . 5 class ↑m
12	8, 10, 11	co 7390	. . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13		vz	. . . . . . . . . . 11 setvar 𝑧
14	13	cv 1539	. . . . . . . . . 10 class 𝑧
15		cnmcv 30526	. . . . . . . . . . 11 class normCV
16	9, 15	cfv 6514	. . . . . . . . . 10 class (normCV‘𝑢)
17	14, 16	cfv 6514	. . . . . . . . 9 class ((normCV‘𝑢)‘𝑧)
18		c1 11076	. . . . . . . . 9 class 1
19		cle 11216	. . . . . . . . 9 class ≤
20	17, 18, 19	wbr 5110	. . . . . . . 8 wff ((normCV‘𝑢)‘𝑧) ≤ 1
21		vx	. . . . . . . . . 10 setvar 𝑥
22	21	cv 1539	. . . . . . . . 9 class 𝑥
23	5	cv 1539	. . . . . . . . . . 11 class 𝑡
24	14, 23	cfv 6514	. . . . . . . . . 10 class (𝑡‘𝑧)
25	6, 15	cfv 6514	. . . . . . . . . 10 class (normCV‘𝑤)
26	24, 25	cfv 6514	. . . . . . . . 9 class ((normCV‘𝑤)‘(𝑡‘𝑧))
27	22, 26	wceq 1540	. . . . . . . 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 3054	. . . . . 6 wff ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
30	29, 21	cab 2708	. . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}
31		cxr 11214	. . . . 5 class ℝ*
32		clt 11215	. . . . 5 class <
33	30, 31, 32	csup 9398	. . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )
34	5, 12, 33	cmpt 5191	. . 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 7392	. 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 1540	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  30698