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| Mirrors > Home > MPE Home > Th. List > df-rrx | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| df-rrx | ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | crrx 25417 | . 2 class ℝ^ | |
| 2 | vi | . . 3 setvar 𝑖 | |
| 3 | cvv 3480 | . . 3 class V | |
| 4 | crefld 21622 | . . . . 5 class ℝfld | |
| 5 | 2 | cv 1539 | . . . . 5 class 𝑖 |
| 6 | cfrlm 21766 | . . . . 5 class freeLMod | |
| 7 | 4, 5, 6 | co 7431 | . . . 4 class (ℝfld freeLMod 𝑖) |
| 8 | ctcph 25201 | . . . 4 class toℂPreHil | |
| 9 | 7, 8 | cfv 6561 | . . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖)) |
| 10 | 2, 3, 9 | cmpt 5225 | . 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) |
| 11 | 1, 10 | wceq 1540 | 1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖))) |
| Colors of variables: wff setvar class |
| This definition is referenced by: rrxval 25421 |
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