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Definition df-nmfn 28674
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 ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))
Distinct variable group:   𝑥,𝑡,𝑧

Detailed syntax breakdown of Definition df-nmfn
StepHypRef Expression
1 cnmf 27778 . 2 class normfn
2 vt . . 3 setvar 𝑡
3 cc 9919 . . . 4 class
4 chil 27746 . . . 4 class
5 cmap 7842 . . . 4 class 𝑚
63, 4, 5co 6635 . . 3 class (ℂ ↑𝑚 ℋ)
7 vz . . . . . . . . . 10 setvar 𝑧
87cv 1480 . . . . . . . . 9 class 𝑧
9 cno 27750 . . . . . . . . 9 class norm
108, 9cfv 5876 . . . . . . . 8 class (norm𝑧)
11 c1 9922 . . . . . . . 8 class 1
12 cle 10060 . . . . . . . 8 class
1310, 11, 12wbr 4644 . . . . . . 7 wff (norm𝑧) ≤ 1
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1480 . . . . . . . 8 class 𝑥
162cv 1480 . . . . . . . . . 10 class 𝑡
178, 16cfv 5876 . . . . . . . . 9 class (𝑡𝑧)
18 cabs 13955 . . . . . . . . 9 class abs
1917, 18cfv 5876 . . . . . . . 8 class (abs‘(𝑡𝑧))
2015, 19wceq 1481 . . . . . . 7 wff 𝑥 = (abs‘(𝑡𝑧))
2113, 20wa 384 . . . . . 6 wff ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))
2221, 7, 4wrex 2910 . . . . 5 wff 𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))
2322, 14cab 2606 . . . 4 class {𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}
24 cxr 10058 . . . 4 class *
25 clt 10059 . . . 4 class <
2623, 24, 25csup 8331 . . 3 class sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < )
272, 6, 26cmpt 4720 . 2 class (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))
281, 27wceq 1481 1 wff normfn = (𝑡 ∈ (ℂ ↑𝑚 ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))
Colors of variables: wff setvar class
This definition is referenced by:  nmfnval  28705
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