# Theory Subspace

theory Subspace
imports Vector_Space Set_Algebras
```(*  Title:      HOL/Hahn_Banach/Subspace.thy
Author:     Gertrud Bauer, TU Munich
*)

section ‹Subspaces›

theory Subspace
imports Vector_Space "HOL-Library.Set_Algebras"
begin

subsection ‹Definition›

text ‹
A non-empty subset ‹U› of a vector space ‹V› is a ∗‹subspace› of ‹V›, iff
‹U› is closed under addition and scalar multiplication.
›

locale subspace =
fixes U :: "'a::{minus, plus, zero, uminus} set" and V
assumes non_empty [iff, intro]: "U ≠ {}"
and subset [iff]: "U ⊆ V"
and add_closed [iff]: "x ∈ U ⟹ y ∈ U ⟹ x + y ∈ U"
and mult_closed [iff]: "x ∈ U ⟹ a ⋅ x ∈ U"

notation (symbols)
subspace  (infix "⊴" 50)

declare vectorspace.intro [intro?] subspace.intro [intro?]

lemma subspace_subset [elim]: "U ⊴ V ⟹ U ⊆ V"
by (rule subspace.subset)

lemma (in subspace) subsetD [iff]: "x ∈ U ⟹ x ∈ V"
using subset by blast

lemma subspaceD [elim]: "U ⊴ V ⟹ x ∈ U ⟹ x ∈ V"
by (rule subspace.subsetD)

lemma rev_subspaceD [elim?]: "x ∈ U ⟹ U ⊴ V ⟹ x ∈ V"
by (rule subspace.subsetD)

lemma (in subspace) diff_closed [iff]:
assumes "vectorspace V"
assumes x: "x ∈ U" and y: "y ∈ U"
shows "x - y ∈ U"
proof -
interpret vectorspace V by fact
from x y show ?thesis by (simp add: diff_eq1 negate_eq1)
qed

text ‹
┉
Similar as for linear spaces, the existence of the zero element in every
subspace follows from the non-emptiness of the carrier set and by vector
space laws.
›

lemma (in subspace) zero [intro]:
assumes "vectorspace V"
shows "0 ∈ U"
proof -
interpret V: vectorspace V by fact
have "U ≠ {}" by (rule non_empty)
then obtain x where x: "x ∈ U" by blast
then have "x ∈ V" .. then have "0 = x - x" by simp
also from ‹vectorspace V› x x have "… ∈ U" by (rule diff_closed)
finally show ?thesis .
qed

lemma (in subspace) neg_closed [iff]:
assumes "vectorspace V"
assumes x: "x ∈ U"
shows "- x ∈ U"
proof -
interpret vectorspace V by fact
from x show ?thesis by (simp add: negate_eq1)
qed

text ‹┉ Further derived laws: every subspace is a vector space.›

lemma (in subspace) vectorspace [iff]:
assumes "vectorspace V"
shows "vectorspace U"
proof -
interpret vectorspace V by fact
show ?thesis
proof
show "U ≠ {}" ..
fix x y z assume x: "x ∈ U" and y: "y ∈ U" and z: "z ∈ U"
fix a b :: real
from x y show "x + y ∈ U" by simp
from x show "a ⋅ x ∈ U" by simp
from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
from x y show "x + y = y + x" by (simp add: add_ac)
from x show "x - x = 0" by simp
from x show "0 + x = x" by simp
from x y show "a ⋅ (x + y) = a ⋅ x + a ⋅ y" by (simp add: distrib)
from x show "(a + b) ⋅ x = a ⋅ x + b ⋅ x" by (simp add: distrib)
from x show "(a * b) ⋅ x = a ⋅ b ⋅ x" by (simp add: mult_assoc)
from x show "1 ⋅ x = x" by simp
from x show "- x = - 1 ⋅ x" by (simp add: negate_eq1)
from x y show "x - y = x + - y" by (simp add: diff_eq1)
qed
qed

text ‹The subspace relation is reflexive.›

lemma (in vectorspace) subspace_refl [intro]: "V ⊴ V"
proof
show "V ≠ {}" ..
show "V ⊆ V" ..
next
fix x y assume x: "x ∈ V" and y: "y ∈ V"
fix a :: real
from x y show "x + y ∈ V" by simp
from x show "a ⋅ x ∈ V" by simp
qed

text ‹The subspace relation is transitive.›

lemma (in vectorspace) subspace_trans [trans]:
"U ⊴ V ⟹ V ⊴ W ⟹ U ⊴ W"
proof
assume uv: "U ⊴ V" and vw: "V ⊴ W"
from uv show "U ≠ {}" by (rule subspace.non_empty)
show "U ⊆ W"
proof -
from uv have "U ⊆ V" by (rule subspace.subset)
also from vw have "V ⊆ W" by (rule subspace.subset)
finally show ?thesis .
qed
fix x y assume x: "x ∈ U" and y: "y ∈ U"
from uv and x y show "x + y ∈ U" by (rule subspace.add_closed)
from uv and x show "a ⋅ x ∈ U" for a by (rule subspace.mult_closed)
qed

subsection ‹Linear closure›

text ‹
The ∗‹linear closure› of a vector ‹x› is the set of all scalar multiples of
‹x›.
›

definition lin :: "('a::{minus,plus,zero}) ⇒ 'a set"
where "lin x = {a ⋅ x | a. True}"

lemma linI [intro]: "y = a ⋅ x ⟹ y ∈ lin x"
unfolding lin_def by blast

lemma linI' [iff]: "a ⋅ x ∈ lin x"
unfolding lin_def by blast

lemma linE [elim]:
assumes "x ∈ lin v"
obtains a :: real where "x = a ⋅ v"
using assms unfolding lin_def by blast

text ‹Every vector is contained in its linear closure.›

lemma (in vectorspace) x_lin_x [iff]: "x ∈ V ⟹ x ∈ lin x"
proof -
assume "x ∈ V"
then have "x = 1 ⋅ x" by simp
also have "… ∈ lin x" ..
finally show ?thesis .
qed

lemma (in vectorspace) "0_lin_x" [iff]: "x ∈ V ⟹ 0 ∈ lin x"
proof
assume "x ∈ V"
then show "0 = 0 ⋅ x" by simp
qed

text ‹Any linear closure is a subspace.›

lemma (in vectorspace) lin_subspace [intro]:
assumes x: "x ∈ V"
shows "lin x ⊴ V"
proof
from x show "lin x ≠ {}" by auto
next
show "lin x ⊆ V"
proof
fix x' assume "x' ∈ lin x"
then obtain a where "x' = a ⋅ x" ..
with x show "x' ∈ V" by simp
qed
next
fix x' x'' assume x': "x' ∈ lin x" and x'': "x'' ∈ lin x"
show "x' + x'' ∈ lin x"
proof -
from x' obtain a' where "x' = a' ⋅ x" ..
moreover from x'' obtain a'' where "x'' = a'' ⋅ x" ..
ultimately have "x' + x'' = (a' + a'') ⋅ x"
using x by (simp add: distrib)
also have "… ∈ lin x" ..
finally show ?thesis .
qed
fix a :: real
show "a ⋅ x' ∈ lin x"
proof -
from x' obtain a' where "x' = a' ⋅ x" ..
with x have "a ⋅ x' = (a * a') ⋅ x" by (simp add: mult_assoc)
also have "… ∈ lin x" ..
finally show ?thesis .
qed
qed

text ‹Any linear closure is a vector space.›

lemma (in vectorspace) lin_vectorspace [intro]:
assumes "x ∈ V"
shows "vectorspace (lin x)"
proof -
from ‹x ∈ V› have "subspace (lin x) V"
by (rule lin_subspace)
from this and vectorspace_axioms show ?thesis
by (rule subspace.vectorspace)
qed

subsection ‹Sum of two vectorspaces›

text ‹
The ∗‹sum› of two vectorspaces ‹U› and ‹V› is the set of all sums of
elements from ‹U› and ‹V›.
›

lemma sum_def: "U + V = {u + v | u v. u ∈ U ∧ v ∈ V}"
unfolding set_plus_def by auto

lemma sumE [elim]:
"x ∈ U + V ⟹ (⋀u v. x = u + v ⟹ u ∈ U ⟹ v ∈ V ⟹ C) ⟹ C"
unfolding sum_def by blast

lemma sumI [intro]:
"u ∈ U ⟹ v ∈ V ⟹ x = u + v ⟹ x ∈ U + V"
unfolding sum_def by blast

lemma sumI' [intro]:
"u ∈ U ⟹ v ∈ V ⟹ u + v ∈ U + V"
unfolding sum_def by blast

text ‹‹U› is a subspace of ‹U + V›.›

lemma subspace_sum1 [iff]:
assumes "vectorspace U" "vectorspace V"
shows "U ⊴ U + V"
proof -
interpret vectorspace U by fact
interpret vectorspace V by fact
show ?thesis
proof
show "U ≠ {}" ..
show "U ⊆ U + V"
proof
fix x assume x: "x ∈ U"
moreover have "0 ∈ V" ..
ultimately have "x + 0 ∈ U + V" ..
with x show "x ∈ U + V" by simp
qed
fix x y assume x: "x ∈ U" and "y ∈ U"
then show "x + y ∈ U" by simp
from x show "a ⋅ x ∈ U" for a by simp
qed
qed

text ‹The sum of two subspaces is again a subspace.›

lemma sum_subspace [intro?]:
assumes "subspace U E" "vectorspace E" "subspace V E"
shows "U + V ⊴ E"
proof -
interpret subspace U E by fact
interpret vectorspace E by fact
interpret subspace V E by fact
show ?thesis
proof
have "0 ∈ U + V"
proof
show "0 ∈ U" using ‹vectorspace E› ..
show "0 ∈ V" using ‹vectorspace E› ..
show "(0::'a) = 0 + 0" by simp
qed
then show "U + V ≠ {}" by blast
show "U + V ⊆ E"
proof
fix x assume "x ∈ U + V"
then obtain u v where "x = u + v" and
"u ∈ U" and "v ∈ V" ..
then show "x ∈ E" by simp
qed
next
fix x y assume x: "x ∈ U + V" and y: "y ∈ U + V"
show "x + y ∈ U + V"
proof -
from x obtain ux vx where "x = ux + vx" and "ux ∈ U" and "vx ∈ V" ..
moreover
from y obtain uy vy where "y = uy + vy" and "uy ∈ U" and "vy ∈ V" ..
ultimately
have "ux + uy ∈ U"
and "vx + vy ∈ V"
and "x + y = (ux + uy) + (vx + vy)"
then show ?thesis ..
qed
fix a show "a ⋅ x ∈ U + V"
proof -
from x obtain u v where "x = u + v" and "u ∈ U" and "v ∈ V" ..
then have "a ⋅ u ∈ U" and "a ⋅ v ∈ V"
and "a ⋅ x = (a ⋅ u) + (a ⋅ v)" by (simp_all add: distrib)
then show ?thesis ..
qed
qed
qed

text ‹The sum of two subspaces is a vectorspace.›

lemma sum_vs [intro?]:
"U ⊴ E ⟹ V ⊴ E ⟹ vectorspace E ⟹ vectorspace (U + V)"
by (rule subspace.vectorspace) (rule sum_subspace)

subsection ‹Direct sums›

text ‹
The sum of ‹U› and ‹V› is called ∗‹direct›, iff the zero element is the only
common element of ‹U› and ‹V›. For every element ‹x› of the direct sum of
‹U› and ‹V› the decomposition in ‹x = u + v› with ‹u ∈ U› and ‹v ∈ V› is
unique.
›

lemma decomp:
assumes "vectorspace E" "subspace U E" "subspace V E"
assumes direct: "U ∩ V = {0}"
and u1: "u1 ∈ U" and u2: "u2 ∈ U"
and v1: "v1 ∈ V" and v2: "v2 ∈ V"
and sum: "u1 + v1 = u2 + v2"
shows "u1 = u2 ∧ v1 = v2"
proof -
interpret vectorspace E by fact
interpret subspace U E by fact
interpret subspace V E by fact
show ?thesis
proof
have U: "vectorspace U"  (* FIXME: use interpret *)
using ‹subspace U E› ‹vectorspace E› by (rule subspace.vectorspace)
have V: "vectorspace V"
using ‹subspace V E› ‹vectorspace E› by (rule subspace.vectorspace)
from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
from u1 u2 have u: "u1 - u2 ∈ U"
by (rule vectorspace.diff_closed [OF U])
with eq have v': "v2 - v1 ∈ U" by (simp only:)
from v2 v1 have v: "v2 - v1 ∈ V"
by (rule vectorspace.diff_closed [OF V])
with eq have u': " u1 - u2 ∈ V" by (simp only:)

show "u1 = u2"
from u1 show "u1 ∈ E" ..
from u2 show "u2 ∈ E" ..
from u u' and direct show "u1 - u2 = 0" by blast
qed
show "v1 = v2"
from v1 show "v1 ∈ E" ..
from v2 show "v2 ∈ E" ..
from v v' and direct show "v2 - v1 = 0" by blast
qed
qed
qed

text ‹
An application of the previous lemma will be used in the proof of the
Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any element
‹y + a ⋅ x⇩0› of the direct sum of a vectorspace ‹H› and the linear closure
of ‹x⇩0› the components ‹y ∈ H› and ‹a› are uniquely determined.
›

lemma decomp_H':
assumes "vectorspace E" "subspace H E"
assumes y1: "y1 ∈ H" and y2: "y2 ∈ H"
and x': "x' ∉ H"  "x' ∈ E"  "x' ≠ 0"
and eq: "y1 + a1 ⋅ x' = y2 + a2 ⋅ x'"
shows "y1 = y2 ∧ a1 = a2"
proof -
interpret vectorspace E by fact
interpret subspace H E by fact
show ?thesis
proof
have c: "y1 = y2 ∧ a1 ⋅ x' = a2 ⋅ x'"
proof (rule decomp)
show "a1 ⋅ x' ∈ lin x'" ..
show "a2 ⋅ x' ∈ lin x'" ..
show "H ∩ lin x' = {0}"
proof
show "H ∩ lin x' ⊆ {0}"
proof
fix x assume x: "x ∈ H ∩ lin x'"
then obtain a where xx': "x = a ⋅ x'"
by blast
have "x = 0"
proof cases
assume "a = 0"
with xx' and x' show ?thesis by simp
next
assume a: "a ≠ 0"
from x have "x ∈ H" ..
with xx' have "inverse a ⋅ a ⋅ x' ∈ H" by simp
with a and x' have "x' ∈ H" by (simp add: mult_assoc2)
with ‹x' ∉ H› show ?thesis by contradiction
qed
then show "x ∈ {0}" ..
qed
show "{0} ⊆ H ∩ lin x'"
proof -
have "0 ∈ H" using ‹vectorspace E› ..
moreover have "0 ∈ lin x'" using ‹x' ∈ E› ..
ultimately show ?thesis by blast
qed
qed
show "lin x' ⊴ E" using ‹x' ∈ E› ..
qed (rule ‹vectorspace E›, rule ‹subspace H E›, rule y1, rule y2, rule eq)
then show "y1 = y2" ..
from c have "a1 ⋅ x' = a2 ⋅ x'" ..
with x' show "a1 = a2" by (simp add: mult_right_cancel)
qed
qed

text ‹
Since for any element ‹y + a ⋅ x'› of the direct sum of a vectorspace ‹H›
and the linear closure of ‹x'› the components ‹y ∈ H› and ‹a› are unique, it
follows from ‹y ∈ H› that ‹a = 0›.
›

lemma decomp_H'_H:
assumes "vectorspace E" "subspace H E"
assumes t: "t ∈ H"
and x': "x' ∉ H"  "x' ∈ E"  "x' ≠ 0"
shows "(SOME (y, a). t = y + a ⋅ x' ∧ y ∈ H) = (t, 0)"
proof -
interpret vectorspace E by fact
interpret subspace H E by fact
show ?thesis
proof (rule, simp_all only: split_paired_all split_conv)
from t x' show "t = t + 0 ⋅ x' ∧ t ∈ H" by simp
fix y and a assume ya: "t = y + a ⋅ x' ∧ y ∈ H"
have "y = t ∧ a = 0"
proof (rule decomp_H')
from ya x' show "y + a ⋅ x' = t + 0 ⋅ x'" by simp
from ya show "y ∈ H" ..
qed (rule ‹vectorspace E›, rule ‹subspace H E›, rule t, (rule x')+)
with t x' show "(y, a) = (y + a ⋅ x', 0)" by simp
qed
qed

text ‹
The components ‹y ∈ H› and ‹a› in ‹y + a ⋅ x'› are unique, so the function
‹h'› defined by ‹h' (y + a ⋅ x') = h y + a ⋅ ξ› is definite.
›

lemma h'_definite:
fixes H
assumes h'_def:
"⋀x. h' x =
(let (y, a) = SOME (y, a). (x = y + a ⋅ x' ∧ y ∈ H)
in (h y) + a * xi)"
and x: "x = y + a ⋅ x'"
assumes "vectorspace E" "subspace H E"
assumes y: "y ∈ H"
and x': "x' ∉ H"  "x' ∈ E"  "x' ≠ 0"
shows "h' x = h y + a * xi"
proof -
interpret vectorspace E by fact
interpret subspace H E by fact
from x y x' have "x ∈ H + lin x'" by auto
have "∃!(y, a). x = y + a ⋅ x' ∧ y ∈ H" (is "∃!p. ?P p")
proof (rule ex_ex1I)
from x y show "∃p. ?P p" by blast
fix p q assume p: "?P p" and q: "?P q"
show "p = q"
proof -
from p have xp: "x = fst p + snd p ⋅ x' ∧ fst p ∈ H"
by (cases p) simp
from q have xq: "x = fst q + snd q ⋅ x' ∧ fst q ∈ H"
by (cases q) simp
have "fst p = fst q ∧ snd p = snd q"
proof (rule decomp_H')
from xp show "fst p ∈ H" ..
from xq show "fst q ∈ H" ..
from xp and xq show "fst p + snd p ⋅ x' = fst q + snd q ⋅ x'"
by simp
qed (rule ‹vectorspace E›, rule ‹subspace H E›, (rule x')+)
then show ?thesis by (cases p, cases q) simp
qed
qed
then have eq: "(SOME (y, a). x = y + a ⋅ x' ∧ y ∈ H) = (y, a)"
by (rule some1_equality) (simp add: x y)
with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
qed

end
```