Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 2,298 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import linear_algebra.affine_space.affine_map
import topology.algebra.group
import topology.algebra.mul_action
/-!
# Topological properties of affine spaces and maps
For now, this contains only a few facts regarding the continuity of affine maps in the special
case when the point space and vector space are the same.
TODO: Deal with the case where the point spaces are different from the vector spaces. Note that
we do have some results in this direction under the assumption that the topologies are induced by
(semi)norms.
-/
namespace affine_map
variables {R E F : Type*}
variables [add_comm_group E] [topological_space E]
variables [add_comm_group F] [topological_space F] [topological_add_group F]
section ring
variables [ring R] [module R E] [module R F]
/-- An affine map is continuous iff its underlying linear map is continuous. See also
`affine_map.continuous_linear_iff`. -/
lemma continuous_iff {f : E →ᵃ[R] F} :
continuous f ↔ continuous f.linear :=
begin
split,
{ intro hc,
rw decomp' f,
have := hc.sub continuous_const,
exact this, },
{ intro hc,
rw decomp f,
have := hc.add continuous_const,
exact this }
end
/-- The line map is continuous. -/
@[continuity]
lemma line_map_continuous [topological_space R] [has_continuous_smul R F] {p v : F} :
continuous ⇑(line_map p v : R →ᵃ[R] F) :=
continuous_iff.mpr $ (continuous_id.smul continuous_const).add $
@continuous_const _ _ _ _ (0 : F)
end ring
section comm_ring
variables [comm_ring R] [module R F] [has_continuous_const_smul R F]
@[continuity]
lemma homothety_continuous (x : F) (t : R) : continuous $ homothety x t :=
begin
suffices : ⇑(homothety x t) = λ y, t • (y - x) + x, { rw this, continuity, },
ext y,
simp [homothety_apply],
end
end comm_ring
section field
variables [field R] [module R F] [has_continuous_const_smul R F]
lemma homothety_is_open_map (x : F) (t : R) (ht : t ≠ 0) : is_open_map $ homothety x t :=
begin
apply is_open_map.of_inverse (homothety_continuous x t⁻¹);
intros e;
simp [← affine_map.comp_apply, ← homothety_mul, ht],
end
end field
end affine_map
|