Definition df-blo 28613

Description: Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
df-blo	⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})

Distinct variable group:   𝑢,𝑡,𝑤

Detailed syntax breakdown of Definition df-blo

Step	Hyp	Ref	Expression
1		cblo 28609	. 2 class BLnOp
2		vu	. . 3 setvar 𝑢
3		vw	. . 3 setvar 𝑤
4		cnv 28451	. . 3 class NrmCVec
5		vt	. . . . . . 7 setvar 𝑡
6	5	cv 1538	. . . . . 6 class 𝑡
7	2	cv 1538	. . . . . . 7 class 𝑢
8	3	cv 1538	. . . . . . 7 class 𝑤
9		cnmoo 28608	. . . . . . 7 class normOpOLD
10	7, 8, 9	co 7143	. . . . . 6 class (𝑢 normOpOLD 𝑤)
11	6, 10	cfv 6328	. . . . 5 class ((𝑢 normOpOLD 𝑤)‘𝑡)
12		cpnf 10695	. . . . 5 class +∞
13		clt 10698	. . . . 5 class <
14	11, 12, 13	wbr 5025	. . . 4 wff ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞
15		clno 28607	. . . . 5 class LnOp
16	7, 8, 15	co 7143	. . . 4 class (𝑢 LnOp 𝑤)
17	14, 5, 16	crab 3072	. . 3 class {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}
18	2, 3, 4, 4, 17	cmpo 7145	. 2 class (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})
19	1, 18	wceq 1539	1 wff BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞})

This definition is referenced by:  bloval  28648