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Definition df-rrx 24831
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 24829 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3473 . . 3 class V
4 crefld 21090 . . . . 5 class fld
52cv 1540 . . . . 5 class 𝑖
6 cfrlm 21234 . . . . 5 class freeLMod
74, 5, 6co 7393 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctcph 24613 . . . 4 class toℂPreHil
97, 8cfv 6532 . . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 5224 . 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1541 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  24833
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