English
If f is a linear map from a finite product, then it is continuous in the product topology, by coordinatewise continuity.
Русский
Линейное отображение из конечного произведения непрерывно по произведенной топологии, координатно непрерывно.
LaTeX
$$$\text{Continuous }(f) \text{ for } f: (\iota \to R) \to M$ with finite ι.$$
Lean4
/-- Constructs a bundled linear map from a function and a proof that this function belongs to the
closure of the set of linear maps. -/
@[simps -fullyApplied]
def linearMapOfMemClosureRangeCoe (f : M₁ → M₂) (hf : f ∈ closure (Set.range ((↑) : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂))) :
M₁ →ₛₗ[σ] M₂ :=
{ addMonoidHomOfMemClosureRangeCoe f hf with
map_smul' :=
(isClosed_setOf_map_smul M₁ M₂ σ).closure_subset_iff.2 (Set.range_subset_iff.2 LinearMap.map_smulₛₗ) hf }
