English
Given a group G acting on a topological module M by linear automorphisms, the action can be encoded as a monoid hom from G to the group of continuous linear automorphisms of M.
Русский
Группа G действует на топологическом модуле M через линейные автоморфизмы; такая дія задает гомоморфизм моноида G в группу непрерывно-линейных автоморфизмов M.
LaTeX
$$$smulLeft : G \\to* (M \\simeq_L[R] M)$$$
Lean4
/-- The product `Π t : α, f t` of a family of topological modules is isomorphic
(both topologically and algebraically) to the space `f ⬝` when `α` only contains `⬝`.
This is `Equiv.piUnique` as a `ContinuousLinearEquiv`.
-/
@[simps! -fullyApplied]
def piUnique {α : Type*} [Unique α] (R : Type*) [Semiring R] (f : α → Type*) [∀ x, AddCommMonoid (f x)]
[∀ x, Module R (f x)] [∀ x, TopologicalSpace (f x)] : (Π t, f t) ≃L[R] f default
where
__ := LinearEquiv.piUnique R f
__ := Homeomorph.piUnique f
