Definition df-nmfn 31774

Description: Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-nmfn	⊢ normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))

Distinct variable group:   𝑥,𝑡,𝑧

Detailed syntax breakdown of Definition df-nmfn

Step	Hyp	Ref	Expression
1		cnmf 30880	. 2 class normfn
2		vt	. . 3 setvar 𝑡
3		cc 11066	. . . 4 class ℂ
4		chba 30848	. . . 4 class ℋ
5		cmap 8799	. . . 4 class ↑m
6	3, 4, 5	co 7387	. . 3 class (ℂ ↑m ℋ)
7		vz	. . . . . . . . . 10 setvar 𝑧
8	7	cv 1539	. . . . . . . . 9 class 𝑧
9		cno 30852	. . . . . . . . 9 class normℎ
10	8, 9	cfv 6511	. . . . . . . 8 class (normℎ‘𝑧)
11		c1 11069	. . . . . . . 8 class 1
12		cle 11209	. . . . . . . 8 class ≤
13	10, 11, 12	wbr 5107	. . . . . . 7 wff (normℎ‘𝑧) ≤ 1
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1539	. . . . . . . 8 class 𝑥
16	2	cv 1539	. . . . . . . . . 10 class 𝑡
17	8, 16	cfv 6511	. . . . . . . . 9 class (𝑡‘𝑧)
18		cabs 15200	. . . . . . . . 9 class abs
19	17, 18	cfv 6511	. . . . . . . 8 class (abs‘(𝑡‘𝑧))
20	15, 19	wceq 1540	. . . . . . 7 wff 𝑥 = (abs‘(𝑡‘𝑧))
21	13, 20	wa 395	. . . . . 6 wff ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))
22	21, 7, 4	wrex 3053	. . . . 5 wff ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))
23	22, 14	cab 2707	. . . 4 class {𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}
24		cxr 11207	. . . 4 class ℝ*
25		clt 11208	. . . 4 class <
26	23, 24, 25	csup 9391	. . 3 class sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < )
27	2, 6, 26	cmpt 5188	. 2 class (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))
28	1, 27	wceq 1540	1 wff normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < ))

This definition is referenced by:  nmfnval  31805