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 24558
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 24556 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3433 . . 3 class V
4 crefld 20818 . . . . 5 class fld
52cv 1538 . . . . 5 class 𝑖
6 cfrlm 20962 . . . . 5 class freeLMod
74, 5, 6co 7284 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctcph 24340 . . . 4 class toℂPreHil
97, 8cfv 6437 . . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 5158 . 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1539 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  24560
  Copyright terms: Public domain W3C validator