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Definition df-rrx 22898
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 22896 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3172 . . 3 class V
4 crefld 19714 . . . . 5 class fld
52cv 1473 . . . . 5 class 𝑖
6 cfrlm 19851 . . . . 5 class freeLMod
74, 5, 6co 6527 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctch 22699 . . . 4 class toℂHil
97, 8cfv 5790 . . 3 class (toℂHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 4637 . 2 class (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1474 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  22900
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