mathlib documentation

topology.continuous_function.units

Units of continuous functions #

This file concerns itself with C(X, M)ˣ and C(X, Mˣ) when X is a topological space and M has some monoid structure compatible with its topology.

Equivalence between continuous maps into the additive units of an additive monoid with continuous addition and the additive units of the additive monoid of continuous maps.

Equations
@[simp]
@[simp]
def continuous_map.units_lift {X : Type u_1} {M : Type u_2} [topological_space X] [monoid M] [topological_space M] [has_continuous_mul M] :

Equivalence between continuous maps into the units of a monoid with continuous multiplication and the units of the monoid of continuous maps.

Equations
theorem normed_ring.is_unit_unit_continuous {X : Type u_1} {R : Type u_3} [topological_space X] [normed_ring R] [complete_space R] {f : C(X, R)} (h : ∀ (x : X), is_unit (f x)) :
continuous (λ (x : X), _.unit)
@[simp]
theorem continuous_map.units_of_forall_is_unit_to_fun {X : Type u_1} {R : Type u_3} [topological_space X] [normed_ring R] [complete_space R] {f : C(X, R)} (h : ∀ (x : X), is_unit (f x)) (x : X) :
noncomputable def continuous_map.units_of_forall_is_unit {X : Type u_1} {R : Type u_3} [topological_space X] [normed_ring R] [complete_space R] {f : C(X, R)} (h : ∀ (x : X), is_unit (f x)) :

Construct a continuous map into the group of units of a normed ring from a function into the normed ring and a proof that every element of the range is a unit.

Equations
@[protected, instance]
def continuous_map.can_lift {X : Type u_1} {R : Type u_3} [topological_space X] [normed_ring R] [complete_space R] :
Equations
theorem continuous_map.is_unit_iff_forall_is_unit {X : Type u_1} {R : Type u_3} [topological_space X] [normed_ring R] [complete_space R] (f : C(X, R)) :
is_unit f ∀ (x : X), is_unit (f x)
theorem continuous_map.is_unit_iff_forall_ne_zero {X : Type u_1} {𝕜 : Type u_4} [topological_space X] [normed_field 𝕜] [complete_space 𝕜] (f : C(X, 𝕜)) :
is_unit f ∀ (x : X), f x 0
theorem continuous_map.spectrum_eq_range {X : Type u_1} {𝕜 : Type u_4} [topological_space X] [normed_field 𝕜] [complete_space 𝕜] (f : C(X, 𝕜)) :