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Definition df-hfmul 27759
Description: Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-hfmul ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hfmul
StepHypRef Expression
1 chft 26965 . 2 class ·fn
2 vf . . 3 setvar 𝑓
3 vg . . 3 setvar 𝑔
4 cc 9693 . . 3 class
5 chil 26942 . . . 4 class
6 cmap 7624 . . . 4 class 𝑚
74, 5, 6co 6431 . . 3 class (ℂ ↑𝑚 ℋ)
8 vx . . . 4 setvar 𝑥
92cv 1473 . . . . 5 class 𝑓
108cv 1473 . . . . . 6 class 𝑥
113cv 1473 . . . . . 6 class 𝑔
1210, 11cfv 5694 . . . . 5 class (𝑔𝑥)
13 cmul 9700 . . . . 5 class ·
149, 12, 13co 6431 . . . 4 class (𝑓 · (𝑔𝑥))
158, 5, 14cmpt 4541 . . 3 class (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥)))
162, 3, 4, 7, 15cmpt2 6433 . 2 class (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
171, 16wceq 1474 1 wff ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))
Colors of variables: wff setvar class
This definition is referenced by:  hfmmval  27764
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