![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ β^ = (π β V β¦ (toβPreHilβ(βfld freeLMod π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crrx 24900 | . 2 class β^ | |
2 | vi | . . 3 setvar π | |
3 | cvv 3475 | . . 3 class V | |
4 | crefld 21157 | . . . . 5 class βfld | |
5 | 2 | cv 1541 | . . . . 5 class π |
6 | cfrlm 21301 | . . . . 5 class freeLMod | |
7 | 4, 5, 6 | co 7409 | . . . 4 class (βfld freeLMod π) |
8 | ctcph 24684 | . . . 4 class toβPreHil | |
9 | 7, 8 | cfv 6544 | . . 3 class (toβPreHilβ(βfld freeLMod π)) |
10 | 2, 3, 9 | cmpt 5232 | . 2 class (π β V β¦ (toβPreHilβ(βfld freeLMod π))) |
11 | 1, 10 | wceq 1542 | 1 wff β^ = (π β V β¦ (toβPreHilβ(βfld freeLMod π))) |
Colors of variables: wff setvar class |
This definition is referenced by: rrxval 24904 |
Copyright terms: Public domain | W3C validator |