Definition df-rrx 25733

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

Step	Hyp	Ref	Expression
1		crrx 25731	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3454	. . 3 class V
4		crefld 21937	. . . . 5 class ℝfld
5	2	cv 1569	. . . . 5 class 𝑖
6		cfrlm 22079	. . . . 5 class freeLMod
7	4, 5, 6	co 7440	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctcph 25515	. . . 4 class toℂPreHil
9	7, 8	cfv 6555	. . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 5189	. 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1570	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

This definition is used by:  rrxval  25735