Metamath Proof Explorer

Definition df-nmoo

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.)

Ref Expression
Assertion df-nmoo ⊢ normOp OLD = u ∈ NrmCVec , w ∈ NrmCVec ⟼ t ∈ BaseSet ⁡ w BaseSet ⁡ u ⟼ sup x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z ℝ * <

Detailed syntax breakdown

Step Hyp Ref Expression
0 cnmoo class normOp OLD
1 vu setvar u
2 cnv class NrmCVec
3 vw setvar w
4 vt setvar t
5 cba class BaseSet
6 3 cv setvar w
7 6 5 cfv class BaseSet ⁡ w
8 cmap class ↑ 𝑚
9 1 cv setvar u
10 9 5 cfv class BaseSet ⁡ u
11 7 10 8 co class BaseSet ⁡ w BaseSet ⁡ u
12 vx setvar x
13 vz setvar z
14 cnmcv class norm CV
15 9 14 cfv class norm CV ⁡ u
16 13 cv setvar z
17 16 15 cfv class norm CV ⁡ u ⁡ z
18 cle class ≤
19 c1 class 1
20 17 19 18 wbr wff norm CV ⁡ u ⁡ z ≤ 1
21 12 cv setvar x
22 6 14 cfv class norm CV ⁡ w
23 4 cv setvar t
24 16 23 cfv class t ⁡ z
25 24 22 cfv class norm CV ⁡ w ⁡ t ⁡ z
26 21 25 wceq wff x = norm CV ⁡ w ⁡ t ⁡ z
27 20 26 wa wff norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z
28 27 13 10 wrex wff ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z
29 28 12 cab class x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z
30 cxr class ℝ *
31 clt class <
32 29 30 31 csup class sup x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z ℝ * <
33 4 11 32 cmpt class t ∈ BaseSet ⁡ w BaseSet ⁡ u ⟼ sup x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z ℝ * <
34 1 3 2 2 33 cmpo class u ∈ NrmCVec , w ∈ NrmCVec ⟼ t ∈ BaseSet ⁡ w BaseSet ⁡ u ⟼ sup x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z ℝ * <
35 0 34 wceq wff normOp OLD = u ∈ NrmCVec , w ∈ NrmCVec ⟼ t ∈ BaseSet ⁡ w BaseSet ⁡ u ⟼ sup x | ∃ z ∈ BaseSet ⁡ u norm CV ⁡ u ⁡ z ≤ 1 ∧ x = norm CV ⁡ w ⁡ t ⁡ z ℝ * <