MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-nmoo Structured version   Visualization version   GIF version

Definition df-nmoo 30774
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 30770 . 2 class normOpOLD
2 vu . . 3 setvar 𝑢
3 vw . . 3 setvar 𝑤
4 cnv 30613 . . 3 class NrmCVec
5 vt . . . 4 setvar 𝑡
63cv 1536 . . . . . 6 class 𝑤
7 cba 30615 . . . . . 6 class BaseSet
86, 7cfv 6563 . . . . 5 class (BaseSet‘𝑤)
92cv 1536 . . . . . 6 class 𝑢
109, 7cfv 6563 . . . . 5 class (BaseSet‘𝑢)
11 cmap 8865 . . . . 5 class m
128, 10, 11co 7431 . . . 4 class ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢))
13 vz . . . . . . . . . . 11 setvar 𝑧
1413cv 1536 . . . . . . . . . 10 class 𝑧
15 cnmcv 30619 . . . . . . . . . . 11 class normCV
169, 15cfv 6563 . . . . . . . . . 10 class (normCV𝑢)
1714, 16cfv 6563 . . . . . . . . 9 class ((normCV𝑢)‘𝑧)
18 c1 11154 . . . . . . . . 9 class 1
19 cle 11294 . . . . . . . . 9 class
2017, 18, 19wbr 5148 . . . . . . . 8 wff ((normCV𝑢)‘𝑧) ≤ 1
21 vx . . . . . . . . . 10 setvar 𝑥
2221cv 1536 . . . . . . . . 9 class 𝑥
235cv 1536 . . . . . . . . . . 11 class 𝑡
2414, 23cfv 6563 . . . . . . . . . 10 class (𝑡𝑧)
256, 15cfv 6563 . . . . . . . . . 10 class (normCV𝑤)
2624, 25cfv 6563 . . . . . . . . 9 class ((normCV𝑤)‘(𝑡𝑧))
2722, 26wceq 1537 . . . . . . . 8 wff 𝑥 = ((normCV𝑤)‘(𝑡𝑧))
2820, 27wa 395 . . . . . . 7 wff (((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
2928, 13, 10wrex 3068 . . . . . 6 wff 𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))
3029, 21cab 2712 . . . . 5 class {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}
31 cxr 11292 . . . . 5 class *
32 clt 11293 . . . . 5 class <
3330, 31, 32csup 9478 . . . 4 class sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )
345, 12, 33cmpt 5231 . . 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 1537 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  30791
  Copyright terms: Public domain W3C validator