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Definition df-nmoo 30764
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)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
Distinct variable group:   𝑢,𝑡,𝑤,𝑥,𝑧

Detailed syntax breakdown of Definition df-nmoo
StepHypRef Expression
1 cnmoo 30760 . 2 class normOpOLD
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 30603 . . 3 class NrmCVec
5 vt . . . 4 setvar 𝑡
63cv 1539 . . . . . 6 class 𝑤
7 cba 30605 . . . . . 6 class BaseSet
86, 7cfv 6561 . . . . 5 class (BaseSet‘𝑤)
92cv 1539 . . . . . 6 class 𝑢
109, 7cfv 6561 . . . . 5 class (BaseSet‘𝑢)
11 cmap 8866 . . . . 5 class m
128, 10, 11co 7431 . . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13 vz . . . . . . . . . . 11 setvar 𝑧
1413cv 1539 . . . . . . . . . 10 class 𝑧
15 cnmcv 30609 . . . . . . . . . . 11 class normCV
169, 15cfv 6561 . . . . . . . . . 10 class (normCV𝑢)
1714, 16cfv 6561 . . . . . . . . 9 class ((normCV𝑢)‘𝑧)
18 c1 11156 . . . . . . . . 9 class 1
19 cle 11296 . . . . . . . . 9 class
2017, 18, 19wbr 5143 . . . . . . . 8 wff ((normCV𝑢)‘𝑧) ≤ 1
21 vx . . . . . . . . . 10 setvar 𝑥
2221cv 1539 . . . . . . . . 9 class 𝑥
235cv 1539 . . . . . . . . . . 11 class 𝑡
2414, 23cfv 6561 . . . . . . . . . 10 class (𝑡𝑧)
256, 15cfv 6561 . . . . . . . . . 10 class (normCV𝑤)
2624, 25cfv 6561 . . . . . . . . 9 class ((normCV𝑤)‘(𝑡𝑧))
2722, 26wceq 1540 . . . . . . . 8 wff 𝑥 = ((normCV𝑤)‘(𝑡𝑧))
2820, 27wa 395 . . . . . . 7 wff (((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
2928, 13, 10wrex 3070 . . . . . 6 wff 𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
3029, 21cab 2714 . . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}
31 cxr 11294 . . . . 5 class *
32 clt 11295 . . . . 5 class <
3330, 31, 32csup 9480 . . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )
345, 12, 33cmpt 5225 . . 3 class (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
352, 3, 4, 4, 34cmpo 7433 . 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
361, 35wceq 1540 1 wff normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
Colors of variables: wff setvar class
This definition is referenced by:  nmoofval  30781
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