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Definition df-nmoo 26984
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 26980 . 2 class normOpOLD
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 26823 . . 3 class NrmCVec
5 vt . . . 4 setvar 𝑡
63cv 1474 . . . . . 6 class 𝑤
7 cba 26825 . . . . . 6 class BaseSet
86, 7cfv 5804 . . . . 5 class (BaseSet‘𝑤)
92cv 1474 . . . . . 6 class 𝑢
109, 7cfv 5804 . . . . 5 class (BaseSet‘𝑢)
11 cmap 7744 . . . . 5 class 𝑚
128, 10, 11co 6549 . . . 4 class ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢))
13 vz . . . . . . . . . . 11 setvar 𝑧
1413cv 1474 . . . . . . . . . 10 class 𝑧
15 cnmcv 26829 . . . . . . . . . . 11 class normCV
169, 15cfv 5804 . . . . . . . . . 10 class (normCV𝑢)
1714, 16cfv 5804 . . . . . . . . 9 class ((normCV𝑢)‘𝑧)
18 c1 9816 . . . . . . . . 9 class 1
19 cle 9954 . . . . . . . . 9 class
2017, 18, 19wbr 4583 . . . . . . . 8 wff ((normCV𝑢)‘𝑧) ≤ 1
21 vx . . . . . . . . . 10 setvar 𝑥
2221cv 1474 . . . . . . . . 9 class 𝑥
235cv 1474 . . . . . . . . . . 11 class 𝑡
2414, 23cfv 5804 . . . . . . . . . 10 class (𝑡𝑧)
256, 15cfv 5804 . . . . . . . . . 10 class (normCV𝑤)
2624, 25cfv 5804 . . . . . . . . 9 class ((normCV𝑤)‘(𝑡𝑧))
2722, 26wceq 1475 . . . . . . . 8 wff 𝑥 = ((normCV𝑤)‘(𝑡𝑧))
2820, 27wa 383 . . . . . . 7 wff (((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
2928, 13, 10wrex 2897 . . . . . 6 wff 𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
3029, 21cab 2596 . . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}
31 cxr 9952 . . . . 5 class *
32 clt 9953 . . . . 5 class <
3330, 31, 32csup 8229 . . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )
345, 12, 33cmpt 4643 . . 3 class (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
352, 3, 4, 4, 34cmpt2 6551 . 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 1475 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  27001
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