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 24902
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 ↦ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)))

Detailed syntax breakdown of Definition df-rrx
StepHypRef Expression
1 crrx 24900 . 2 class ℝ^
2 vi . . 3 setvar 𝑖
3 cvv 3475 . . 3 class V
4 crefld 21157 . . . . 5 class ℝfld
52cv 1541 . . . . 5 class 𝑖
6 cfrlm 21301 . . . . 5 class freeLMod
74, 5, 6co 7409 . . . 4 class (ℝfld freeLMod 𝑖)
8 ctcph 24684 . . . 4 class toβ„‚PreHil
97, 8cfv 6544 . . 3 class (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖))
102, 3, 9cmpt 5232 . 2 class (𝑖 ∈ V ↦ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)))
111, 10wceq 1542 1 wff ℝ^ = (𝑖 ∈ V ↦ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)))
Colors of variables: wff setvar class
This definition is referenced by:  rrxval  24904
  Copyright terms: Public domain W3C validator