MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rrx Structured version   Visualization version   GIF version

Definition df-rrx 24137
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 24135 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3398 . . 3 class V
4 crefld 20420 . . . . 5 class fld
52cv 1541 . . . . 5 class 𝑖
6 cfrlm 20562 . . . . 5 class freeLMod
74, 5, 6co 7170 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctcph 23919 . . . 4 class toℂPreHil
97, 8cfv 6339 . . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 5110 . 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1542 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  24139
  Copyright terms: Public domain W3C validator