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Definition df-nmfn 29626
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 28732 . 2 class normfn
2 vt . . 3 setvar 𝑡
3 cc 10524 . . . 4 class
4 chba 28700 . . . 4 class
5 cmap 8393 . . . 4 class m
63, 4, 5co 7140 . . 3 class (ℂ ↑m ℋ)
7 vz . . . . . . . . . 10 setvar 𝑧
87cv 1537 . . . . . . . . 9 class 𝑧
9 cno 28704 . . . . . . . . 9 class norm
108, 9cfv 6334 . . . . . . . 8 class (norm𝑧)
11 c1 10527 . . . . . . . 8 class 1
12 cle 10665 . . . . . . . 8 class
1310, 11, 12wbr 5042 . . . . . . 7 wff (norm𝑧) ≤ 1
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1537 . . . . . . . 8 class 𝑥
162cv 1537 . . . . . . . . . 10 class 𝑡
178, 16cfv 6334 . . . . . . . . 9 class (𝑡𝑧)
18 cabs 14584 . . . . . . . . 9 class abs
1917, 18cfv 6334 . . . . . . . 8 class (abs‘(𝑡𝑧))
2015, 19wceq 1538 . . . . . . 7 wff 𝑥 = (abs‘(𝑡𝑧))
2113, 20wa 399 . . . . . 6 wff ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))
2221, 7, 4wrex 3131 . . . . 5 wff 𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))
2322, 14cab 2800 . . . 4 class {𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}
24 cxr 10663 . . . 4 class *
25 clt 10664 . . . 4 class <
2623, 24, 25csup 8892 . . 3 class sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < )
272, 6, 26cmpt 5122 . 2 class (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))
281, 27wceq 1538 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  29657
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