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/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import topology.algebra.module.weak_dual
import analysis.normed.field.basic
import analysis.locally_convex.with_seminorms
/-!
# Weak Dual in Topological Vector Spaces
We prove that the weak topology induced by a bilinear form `B : E ββ[π] F ββ[π] π` is locally
convex and we explicit give a neighborhood basis in terms of the family of seminorms `Ξ» x, β₯B x yβ₯`
for `y : F`.
## Main definitions
* `linear_map.to_seminorm`: turn a linear form `f : E ββ[π] π` into a seminorm `Ξ» x, β₯f xβ₯`.
* `linear_map.to_seminorm_family`: turn a bilinear form `B : E ββ[π] F ββ[π] π` into a map
`F β seminorm π E`.
## Main statements
* `linear_map.has_basis_weak_bilin`: the seminorm balls of `B.to_seminorm_family` form a
neighborhood basis of `0` in the weak topology.
* `linear_map.to_seminorm_family.with_seminorms`: the topology of a weak space is induced by the
family of seminorm `B.to_seminorm_family`.
* `weak_bilin.locally_convex_space`: a spaced endowed with a weak topology is locally convex.
## References
* [Bourbaki, *Topological Vector Spaces*][bourbaki1987]
## Tags
weak dual, seminorm
-/
variables {π E F ΞΉ : Type*}
open_locale topological_space
section bilin_form
namespace linear_map
variables [normed_field π] [add_comm_group E] [module π E] [add_comm_group F] [module π F]
/-- Construct a seminorm from a linear form `f : E ββ[π] π` over a normed field `π` by
`Ξ» x, β₯f xβ₯` -/
def to_seminorm (f : E ββ[π] π) : seminorm π E :=
(norm_seminorm π π).comp f
lemma coe_to_seminorm {f : E ββ[π] π} :
βf.to_seminorm = Ξ» x, β₯f xβ₯ := rfl
@[simp] lemma to_seminorm_apply {f : E ββ[π] π} {x : E} :
f.to_seminorm x = β₯f xβ₯ := rfl
lemma to_seminorm_ball_zero {f : E ββ[π] π} {r : β} :
seminorm.ball f.to_seminorm 0 r = { x : E | β₯f xβ₯ < r} :=
by simp only [seminorm.ball_zero_eq, to_seminorm_apply]
lemma to_seminorm_comp (f : F ββ[π] π) (g : E ββ[π] F) :
f.to_seminorm.comp g = (f.comp g).to_seminorm :=
by { ext, simp only [seminorm.comp_apply, to_seminorm_apply, coe_comp] }
/-- Construct a family of seminorms from a bilinear form. -/
def to_seminorm_family (B : E ββ[π] F ββ[π] π) : seminorm_family π E F :=
Ξ» y, (B.flip y).to_seminorm
@[simp] lemma to_seminorm_family_apply {B : E ββ[π] F ββ[π] π} {x y} :
(B.to_seminorm_family y) x = β₯B x yβ₯ := rfl
end linear_map
end bilin_form
section topology
variables [normed_field π] [add_comm_group E] [module π E] [add_comm_group F] [module π F]
variables [nonempty ΞΉ]
variables {B : E ββ[π] F ββ[π] π}
lemma linear_map.has_basis_weak_bilin (B : E ββ[π] F ββ[π] π) :
(π (0 : weak_bilin B)).has_basis B.to_seminorm_family.basis_sets id :=
begin
let p := B.to_seminorm_family,
rw [nhds_induced, nhds_pi],
simp only [map_zero, linear_map.zero_apply],
have h := @metric.nhds_basis_ball π _ 0,
have h' := filter.has_basis_pi (Ξ» (i : F), h),
have h'' := filter.has_basis.comap (Ξ» x y, B x y) h',
refine h''.to_has_basis _ _,
{ rintros (U : set F Γ (F β β)) hU,
cases hU with hUβ hUβ,
simp only [id.def],
let U' := hUβ.to_finset,
by_cases hUβ : U.fst.nonempty,
{ have hUβ' : U'.nonempty := hUβ.nonempty_to_finset.mpr hUβ,
refine β¨(U'.sup p).ball 0 $ U'.inf' hUβ' U.snd, p.basis_sets_mem _ $
(finset.lt_inf'_iff _).2 $ Ξ» y hy, hUβ y $ (hUβ.mem_to_finset).mp hy, Ξ» x hx y hy, _β©,
simp only [set.mem_preimage, set.mem_pi, mem_ball_zero_iff],
rw seminorm.mem_ball_zero at hx,
rw βlinear_map.to_seminorm_family_apply,
have hyU' : y β U' := (set.finite.mem_to_finset hUβ).mpr hy,
have hp : p y β€ U'.sup p := finset.le_sup hyU',
refine lt_of_le_of_lt (hp x) (lt_of_lt_of_le hx _),
exact finset.inf'_le _ hyU' },
rw set.not_nonempty_iff_eq_empty.mp hUβ,
simp only [set.empty_pi, set.preimage_univ, set.subset_univ, and_true],
exact Exists.intro ((p 0).ball 0 1) (p.basis_sets_singleton_mem 0 one_pos) },
rintros U (hU : U β p.basis_sets),
rw seminorm_family.basis_sets_iff at hU,
rcases hU with β¨s, r, hr, hUβ©,
rw hU,
refine β¨(s, Ξ» _, r), β¨by simp only [s.finite_to_set], Ξ» y hy, hrβ©, Ξ» x hx, _β©,
simp only [set.mem_preimage, set.mem_pi, finset.mem_coe, mem_ball_zero_iff] at hx,
simp only [id.def, seminorm.mem_ball, sub_zero],
refine seminorm.finset_sup_apply_lt hr (Ξ» y hy, _),
rw linear_map.to_seminorm_family_apply,
exact hx y hy,
end
lemma linear_map.weak_bilin_with_seminorms (B : E ββ[π] F ββ[π] π) :
with_seminorms (linear_map.to_seminorm_family B : F β seminorm π (weak_bilin B)) :=
seminorm_family.with_seminorms_of_has_basis _ B.has_basis_weak_bilin
end topology
section locally_convex
variables [normed_field π] [add_comm_group E] [module π E] [add_comm_group F] [module π F]
variables [nonempty ΞΉ] [normed_space β π] [module β E] [is_scalar_tower β π E]
instance {B : E ββ[π] F ββ[π] π} : locally_convex_space β (weak_bilin B) :=
seminorm_family.to_locally_convex_space (B.weak_bilin_with_seminorms)
end locally_convex
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