Definition df-hnorm 8839

Description: Define the function for the norm of a vector of Hilbert space. See normvalt 8992 for its value and normclt 8993 for its closure. Theorems norm-i 9002, norm-ii 9006, and norm-iii 9008 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 8796	. 2 class normh
2		vx	. . . . . 6 set x
3	2	cv 957	. . . . 5 class x
4		csp 8795	. . . . . . 7 class .ih
5	4	cdm 3177	. . . . . 6 class dom .ih
6	5	cdm 3177	. . . . 5 class dom dom .ih
7	3, 6	wcel 960	. . . 4 wff x e. dom dom .ih
8		vy	. . . . . 6 set y
9	8	cv 957	. . . . 5 class y
10	3, 3, 4	co 3970	. . . . . 6 class (x .ih x)
11		csqr 6677	. . . . . 6 class sqr
12	10, 11	cfv 3189	. . . . 5 class (sqr` (x .ih x))
13	9, 12	wceq 958	. . . 4 wff y = (sqr` (x .ih x))
14	7, 13	wa 223	. . 3 wff (x e. dom dom .ih /\ y = (sqr` (x .ih x)))
15	14, 2, 8	copab 2672	. 2 class {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}
16	1, 15	wceq 958	1 wff normh = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}

This definition is referenced by:  dfhnorm2 8990

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