Definition df-blo 8407

Description: Define the class of bounded linear operators between two normed complex vector spaces.

Assertion

Ref	Expression
df-blo	|- BLnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})}

Distinct variable group:   t,o,u,w

Detailed syntax breakdown of Definition df-blo

Step	Hyp	Ref	Expression
1		cblo 8403	. 2 class BLnOp
2		vu	. . . . . . 7 set u
3	2	cv 955	. . . . . 6 class u
4		cnv 8203	. . . . . 6 class NrmCVec
5	3, 4	wcel 958	. . . . 5 wff u e. NrmCVec
6		vw	. . . . . . 7 set w
7	6	cv 955	. . . . . 6 class w
8	7, 4	wcel 958	. . . . 5 wff w e. NrmCVec
9	5, 8	wa 223	. . . 4 wff (u e. NrmCVec /\ w e. NrmCVec)
10		vo	. . . . . 6 set o
11	10	cv 955	. . . . 5 class o
12		vt	. . . . . . . . 9 set t
13	12	cv 955	. . . . . . . 8 class t
14		cnmo 8402	. . . . . . . . 9 class normOp
15	3, 7, 14	co 3963	. . . . . . . 8 class (unormOpw)
16	13, 15	cfv 3182	. . . . . . 7 class ((unormOpw)` t)
17		cpnf 5483	. . . . . . 7 class +oo
18		clt 5486	. . . . . . 7 class <
19	16, 17, 18	wbr 2619	. . . . . 6 wff ((unormOpw)` t) < +oo
20		clno 8401	. . . . . . 7 class LnOp
21	3, 7, 20	co 3963	. . . . . 6 class (u LnOp w)
22	19, 12, 21	crab 1648	. . . . 5 class {t e. (u LnOp w) | ((unormOpw)` t) < +oo}
23	11, 22	wceq 956	. . . 4 wff o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo}
24	9, 23	wa 223	. . 3 wff ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})
25	24, 2, 6, 10	copab2 3964	. 2 class {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})}
26	1, 25	wceq 956	1 wff BLnOp = {<.<.u, w>., o>. | ((u e. NrmCVec /\ w e. NrmCVec) /\ o = {t e. (u LnOp w) | ((unormOpw)` t) < +oo})}

This definition is referenced by:  bloval 8441

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