English
The map completeFamily assigns to each infinite place i a vector in realSpace(K); together with the vector mult w for w0, this family forms a basis of realSpace(K).
Русский
Осуществление completeFamily для каждого бесконечного места i даёт вектор в realSpace(K); вместе с вектором mult w при w0 эта коллекция образует базис пространства реальных величин realSpace(K).
LaTeX
$$$\text{completeFamily}: \mathrm{InfinitePlace}(K) \to \mathrm{realSpace}(K)$ and the family \{\text{completeFamily}(w) \mid w \in \mathrm{InfinitePlace}(K)\} is a basis of $\mathrm{realSpace}(K)$.$$
Lean4
/-- A family of elements in the `realSpace K` formed of the image of the fundamental units
and the vector `(mult w)_w`. This family is in fact a basis of `realSpace K`, see `completeBasis`.
-/
def completeFamily : InfinitePlace K → realSpace K := fun i ↦
if hi : i = w₀ then fun w ↦ mult w
else expMap.symm <| normAtAllPlaces <| mixedEmbedding K <| fundSystem K <| equivFinRank.symm ⟨i, hi⟩
