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 23219
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 23217 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3231 . . 3 class V
4 crefld 19998 . . . . 5 class fld
52cv 1522 . . . . 5 class 𝑖
6 cfrlm 20138 . . . . 5 class freeLMod
74, 5, 6co 6690 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctch 23013 . . . 4 class toℂHil
97, 8cfv 5926 . . 3 class (toℂHil‘(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 4762 . 2 class (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
111, 10wceq 1523 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  23221
  Copyright terms: Public domain W3C validator