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Evt #1107

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41 changes: 39 additions & 2 deletions theories/topology.v
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Original file line number Diff line number Diff line change
Expand Up @@ -4762,7 +4762,45 @@ End sup_uniform.
HB.instance Definition _ (I : Type) (T : I -> uniformType) :=
Uniform.copy (prod_topology T)
(sup_topology (fun i => Uniform.class
[the uniformType of weak_topology (@proj _ T i)])).
[the uniformType of weak_topology (@proj _ T i)])).

(** Uniform of Normed Domain**)


HB.instance Definition _ (R : zmodType) := isPointed.Build R 0.

HB.instance Definition _ (R : zmodType) := Pointed.on R^o.

HB.howto GRing.regular topologicalType.

Definition numDomain_axiom (R : numDomainType) : Pointed_isBaseTopological R^o.
Admitted.
HB.instance Definition _ (R : numDomainType) := numDomain_axiom R.

(** * Topological vector spaces *)

Definition convex (R : numDomainType) (M : lmodType R) (A : set M) :=
forall x y lambda, lambda *: x + (1 - lambda) *: y \in A.

HB.mixin Record Uniform_isTopvec (R : numDomainType) E of Uniform E & GRing.Lmodule R E := {
add_continuous : continuous (uncurry (@GRing.add E));
scale_continuous : continuous (uncurry (@GRing.scale R E) : R^o * E -> E);
locally_convex : exists B : set (set E), (forall b, b \in B -> convex b) /\ (basis B)
}.

HB.structure Definition Evt (R : numDomainType) :=
{E of Uniform_isTopvec R E & Uniform E & GRing.Lmodule R E}.

HB.factory Record Topological_isTopvec (R : numDomainType) E
of Topological E & GRing.Lmodule R E := {
add_continuous : continuous (uncurry (@GRing.add E));
scale_continuous : continuous (uncurry (@GRing.scale R E) : R^o * E -> E);
locally_convex : exists B : set (set E), (forall b, b \in B -> convex b) /\ (basis B)
}.
HB.builders Context R E of Topological_isTopvec R E.

HB.end.


(** * PseudoMetric spaces defined using balls *)

Expand Down Expand Up @@ -5465,7 +5503,6 @@ HB.instance Definition _ (T : choiceType) (R : numFieldType)
(U : completePseudoMetricType R) :=
Uniform_isComplete.Build (T -> U) cauchy_cvg.

HB.instance Definition _ (R : zmodType) := isPointed.Build R 0.

Lemma compact_cauchy_cvg {T : uniformType} (U : set T) (F : set_system T) :
ProperFilter F -> cauchy F -> F U -> compact U -> cvg F.
Expand Down