Definition df-hnorm 27825

Description: Define the function for the norm of a vector of Hilbert space. See normval 27981 for its value and normcl 27982 for its closure. Theorems norm-i-i 27990, norm-ii-i 27994, and norm-iii-i 27996 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 e. dom dom .ih |-> ( sqr ` ( x .ih x ) ) )

Detailed syntax breakdown of Definition df-hnorm

Step	Hyp	Ref	Expression
1		cno 27780	. 2 class normh
2		vx	. . 3 setvar x
3		csp 27779	. . . . 5 class .ih
4	3	cdm 5114	. . . 4 class dom .ih
5	4	cdm 5114	. . 3 class dom dom .ih
6	2	cv 1482	. . . . 5 class x
7	6, 6, 3	co 6650	. . . 4 class ( x .ih x )
8		csqrt 13973	. . . 4 class sqr
9	7, 8	cfv 5888	. . 3 class ( sqr ` ( x .ih x ) )
10	2, 5, 9	cmpt 4729	. 2 class ( x e. dom dom .ih |-> ( sqr ` ( x .ih x ) ) )
11	1, 10	wceq 1483	1 wff normh = ( x e. dom dom .ih |-> ( sqr ` ( x .ih x ) ) )

This definition is referenced by:  dfhnorm2  27979