Definition df-hnorm 9112

Description: Define the function for the norm of a vector of Hilbert space. See normval 9266 for its value and normcl 9267 for its closure. Theorems norm-i.i 9276, norm-ii.i 9280, and norm-iii.i 9282 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.

Assertion

Ref	Expression
df-hnorm	|- normh = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}

Distinct variable group:   x,y

Detailed syntax breakdown of Definition df-hnorm

Step	Hyp	Ref	Expression
1		cno 9069	. 2 class normh
2		vx	. . . . . 6 set x
3	2	cv 991	. . . . 5 class x
4		csp 9068	. . . . . . 7 class .ih
5	4	cdm 3251	. . . . . 6 class dom .ih
6	5	cdm 3251	. . . . 5 class dom dom .ih
7	3, 6	wcel 994	. . . 4 wff x e. dom dom .ih
8		vy	. . . . . 6 set y
9	8	cv 991	. . . . 5 class y
10	3, 3, 4	co 4021	. . . . . 6 class (x .ih x)
11		csqr 6870	. . . . . 6 class sqr
12	10, 11	cfv 3263	. . . . 5 class (sqr` (x .ih x))
13	9, 12	wceq 992	. . . 4 wff y = (sqr` (x .ih x))
14	7, 13	wa 221	. . 3 wff (x e. dom dom .ih /\ y = (sqr` (x .ih x)))
15	14, 2, 8	copab 2740	. 2 class {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}
16	1, 15	wceq 992	1 wff normh = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}

This definition is referenced by:  dfhnorm2 9264

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