Definition df-hnorm 30987

Description: Define the function for the norm of a vector of Hilbert space. See normval 31143 for its value and normcl 31144 for its closure. Theorems norm-i-i 31152, norm-ii-i 31156, and norm-iii-i 31158 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	⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))

Detailed syntax breakdown of Definition df-hnorm

Step	Hyp	Ref	Expression
1		cno 30942	. 2 class normℎ
2		vx	. . 3 setvar 𝑥
3		csp 30941	. . . . 5 class ·ih
4	3	cdm 5685	. . . 4 class dom ·ih
5	4	cdm 5685	. . 3 class dom dom ·ih
6	2	cv 1539	. . . . 5 class 𝑥
7	6, 6, 3	co 7431	. . . 4 class (𝑥 ·ih 𝑥)
8		csqrt 15272	. . . 4 class √
9	7, 8	cfv 6561	. . 3 class (√‘(𝑥 ·ih 𝑥))
10	2, 5, 9	cmpt 5225	. 2 class (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))
11	1, 10	wceq 1540	1 wff normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥)))

This definition is referenced by:  dfhnorm2  31141