Definition df-hosum 10931

Description: Define the sum of two Hilbert space operators. Definition of [Beran] p. 111.

Note on operators. Unlike some authors, we use the term "operator" to mean any function from ~H to ~H. This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 11396.

Assertion

Ref	Expression
df-hosum	|- +op = {<.<.f, g>., h>. | ((f:~H-->~H /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))})}

Distinct variable group:   f,g,h,x,y

Detailed syntax breakdown of Definition df-hosum

Step	Hyp	Ref	Expression
1		chos 10231	. 2 class +op
2		chil 10212	. . . . . 6 class ~H
3		vf	. . . . . . 7 set f
4	3	cv 1135	. . . . . 6 class f
5	2, 2, 4	wf 3805	. . . . 5 wff f:~H-->~H
6		vg	. . . . . . 7 set g
7	6	cv 1135	. . . . . 6 class g
8	2, 2, 7	wf 3805	. . . . 5 wff g:~H-->~H
9	5, 8	wa 239	. . . 4 wff (f:~H-->~H /\ g:~H-->~H)
10		vh	. . . . . 6 set h
11	10	cv 1135	. . . . 5 class h
12		vx	. . . . . . . . 9 set x
13	12	cv 1135	. . . . . . . 8 class x
14	13, 2	wcel 1138	. . . . . . 7 wff x e. ~H
15		vy	. . . . . . . . 9 set y
16	15	cv 1135	. . . . . . . 8 class y
17	13, 4	cfv 3809	. . . . . . . . 9 class (f` x)
18	13, 7	cfv 3809	. . . . . . . . 9 class (g` x)
19		cva 10213	. . . . . . . . 9 class +h
20	17, 18, 19	co 4695	. . . . . . . 8 class ((f` x) +h (g` x))
21	16, 20	wceq 1136	. . . . . . 7 wff y = ((f` x) +h (g` x))
22	14, 21	wa 239	. . . . . 6 wff (x e. ~H /\ y = ((f` x) +h (g` x)))
23	22, 12, 15	copab 3213	. . . . 5 class {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))}
24	11, 23	wceq 1136	. . . 4 wff h = {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))}
25	9, 24	wa 239	. . 3 wff ((f:~H-->~H /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))})
26	25, 3, 6, 10	copab2 4696	. 2 class {<.<.f, g>., h>. | ((f:~H-->~H /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))})}
27	1, 26	wceq 1136	1 wff +op = {<.<.f, g>., h>. | ((f:~H-->~H /\ g:~H-->~H) /\ h = {<.x, y>. | (x e. ~H /\ y = ((f` x) +h (g` x)))})}

This definition is referenced by:  hosmval 10936

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