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Definition df-hfsum 28720
 Description: Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from ℋ to ℂ, not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-hfsum +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hfsum
StepHypRef Expression
1 chfs 27926 . 2 class +fn
2 vf . . 3 setvar 𝑓
3 vg . . 3 setvar 𝑔
4 cc 9972 . . . 4 class
5 chil 27904 . . . 4 class
6 cmap 7899 . . . 4 class 𝑚
74, 5, 6co 6690 . . 3 class (ℂ ↑𝑚 ℋ)
8 vx . . . 4 setvar 𝑥
98cv 1522 . . . . . 6 class 𝑥
102cv 1522 . . . . . 6 class 𝑓
119, 10cfv 5926 . . . . 5 class (𝑓𝑥)
123cv 1522 . . . . . 6 class 𝑔
139, 12cfv 5926 . . . . 5 class (𝑔𝑥)
14 caddc 9977 . . . . 5 class +
1511, 13, 14co 6690 . . . 4 class ((𝑓𝑥) + (𝑔𝑥))
168, 5, 15cmpt 4762 . . 3 class (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥)))
172, 3, 7, 7, 16cmpt2 6692 . 2 class (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
181, 17wceq 1523 1 wff +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
 Colors of variables: wff setvar class This definition is referenced by:  hfsmval  28725
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