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Definition df-homul 28574
 Description: Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-homul ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-homul
StepHypRef Expression
1 chot 27780 . 2 class ·op
2 vf . . 3 setvar 𝑓
3 vg . . 3 setvar 𝑔
4 cc 9931 . . 3 class
5 chil 27760 . . . 4 class
6 cmap 7854 . . . 4 class 𝑚
75, 5, 6co 6647 . . 3 class ( ℋ ↑𝑚 ℋ)
8 vx . . . 4 setvar 𝑥
92cv 1481 . . . . 5 class 𝑓
108cv 1481 . . . . . 6 class 𝑥
113cv 1481 . . . . . 6 class 𝑔
1210, 11cfv 5886 . . . . 5 class (𝑔𝑥)
13 csm 27762 . . . . 5 class ·
149, 12, 13co 6647 . . . 4 class (𝑓 · (𝑔𝑥))
158, 5, 14cmpt 4727 . . 3 class (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥)))
162, 3, 4, 7, 15cmpt2 6649 . 2 class (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
171, 16wceq 1482 1 wff ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
 Colors of variables: wff setvar class This definition is referenced by:  hommval  28579
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