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Definition df-rrx 22981
 Description: Define the function associating with a set the free real vector space on that set, equipped with the natural inner product. This is the direct sum of copies of the field of real numbers indexed by that set. We call it here a "generalized real Euclidean space", but note that it need not be complete (for instance if the given set is infinite countable). (Contributed by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
df-rrx ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))

Detailed syntax breakdown of Definition df-rrx
StepHypRef Expression
1 crrx 22979 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3173 . . 3 class V
4 crefld 19769 . . . . 5 class fld
52cv 1474 . . . . 5 class 𝑖
6 cfrlm 19909 . . . . 5 class freeLMod
74, 5, 6co 6549 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctch 22775 . . . 4 class toℂHil
97, 8cfv 5804 . . 3 class (toℂHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 4643 . 2 class (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1475 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
 Colors of variables: wff setvar class This definition is referenced by:  rrxval  22983
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