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Definition df-rrx 23511
Description: Define the function associating with a set the free real vector space on that set, equipped with the natural inner product and norm. 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 ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

Detailed syntax breakdown of Definition df-rrx
StepHypRef Expression
1 crrx 23509 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3385 . . 3 class V
4 crefld 20273 . . . . 5 class fld
52cv 1652 . . . . 5 class 𝑖
6 cfrlm 20415 . . . . 5 class freeLMod
74, 5, 6co 6878 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctcph 23294 . . . 4 class toℂPreHil
97, 8cfv 6101 . . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 4922 . 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1653 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  23513
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