Definition df-nmoo 30689

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 30685	. 2 class normOpOLD
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 30528	. . 3 class NrmCVec
5		vt	. . . 4 setvar 𝑡
6	3	cv 1539	. . . . . 6 class 𝑤
7		cba 30530	. . . . . 6 class BaseSet
8	6, 7	cfv 6482	. . . . 5 class (BaseSet‘𝑤)
9	2	cv 1539	. . . . . 6 class 𝑢
10	9, 7	cfv 6482	. . . . 5 class (BaseSet‘𝑢)
11		cmap 8753	. . . . 5 class ↑m
12	8, 10, 11	co 7349	. . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13		vz	. . . . . . . . . . 11 setvar 𝑧
14	13	cv 1539	. . . . . . . . . 10 class 𝑧
15		cnmcv 30534	. . . . . . . . . . 11 class normCV
16	9, 15	cfv 6482	. . . . . . . . . 10 class (normCV‘𝑢)
17	14, 16	cfv 6482	. . . . . . . . 9 class ((normCV‘𝑢)‘𝑧)
18		c1 11010	. . . . . . . . 9 class 1
19		cle 11150	. . . . . . . . 9 class ≤
20	17, 18, 19	wbr 5092	. . . . . . . 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 6482	. . . . . . . . . 10 class (𝑡‘𝑧)
25	6, 15	cfv 6482	. . . . . . . . . 10 class (normCV‘𝑤)
26	24, 25	cfv 6482	. . . . . . . . 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 3053	. . . . . 6 wff ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))
30	29, 21	cab 2707	. . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}
31		cxr 11148	. . . . 5 class ℝ*
32		clt 11149	. . . . 5 class <
33	30, 31, 32	csup 9330	. . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )
34	5, 12, 33	cmpt 5173	. . 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 7351	. 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  30706