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Definition df-hnorm 27001
Description: Define the function for the norm of a vector of Hilbert space. See normval 27157 for its value and normcl 27158 for its closure. Theorems norm-i-i 27166, norm-ii-i 27170, and norm-iii-i 27172 show it has the expected properties of a norm. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-hnorm norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))

Detailed syntax breakdown of Definition df-hnorm
StepHypRef Expression
1 cno 26956 . 2 class norm
2 vx . . 3 setvar 𝑥
3 csp 26955 . . . . 5 class ·ih
43cdm 4932 . . . 4 class dom ·ih
54cdm 4932 . . 3 class dom dom ·ih
62cv 1473 . . . . 5 class 𝑥
76, 6, 3co 6431 . . . 4 class (𝑥 ·ih 𝑥)
8 csqrt 13684 . . . 4 class
97, 8cfv 5694 . . 3 class (√‘(𝑥 ·ih 𝑥))
102, 5, 9cmpt 4541 . 2 class (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
111, 10wceq 1474 1 wff norm = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  dfhnorm2  27155
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