(* Title: HOL/Hahn_Banach/Subspace.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Subspaces\<close>
theory Subspace
imports Vector_Space "HOL-Library.Set_Algebras"
begin
subsection \<open>Definition\<close>
text \<open>
A non-empty subset \<open>U\<close> of a vector space \<open>V\<close> is a \<^emph>\<open>subspace\<close> of \<open>V\<close>, iff
\<open>U\<close> is closed under addition and scalar multiplication.
\<close>
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 \<open>
\<^medskip>
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.
\<close>
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 \<open>vectorspace V\<close> x x have "\<dots> \<in> 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 \<open>\<^medskip> Further derived laws: every subspace is a vector space.\<close>
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 \<open>The subspace relation is reflexive.\<close>
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 \<open>The subspace relation is transitive.\<close>
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 \<open>Linear closure\<close>
text \<open>
The \<^emph>\<open>linear closure\<close> of a vector \<open>x\<close> is the set of all scalar multiples of
\<open>x\<close>.
\<close>
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 \<open>Every vector is contained in its linear closure.\<close>
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 \<open>Any linear closure is a subspace.\<close>
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' \<in> 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 \<open>Any linear closure is a vector space.\<close>
lemma (in vectorspace) lin_vectorspace [intro]:
assumes "x \ V"
shows "vectorspace (lin x)"
proof -
from \<open>x \<in> V\<close> have "subspace (lin x) V"
by (rule lin_subspace)
from this and vectorspace_axioms show ?thesis
by (rule subspace.vectorspace)
qed
subsection \<open>Sum of two vectorspaces\<close>
text \<open>
The \<^emph>\<open>sum\<close> of two vectorspaces \<open>U\<close> and \<open>V\<close> is the set of all sums of
elements from \<open>U\<close> and \<open>V\<close>.
\<close>
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 \<open>\<open>U\<close> is a subspace of \<open>U + V\<close>.\<close>
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 \<open>The sum of two subspaces is again a subspace.\<close>
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)"
using x y by (simp_all add: add_ac)
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 \<open>The sum of two subspaces is a vectorspace.\<close>
lemma sum_vs [intro?]:
"U \ E \ V \ E \ vectorspace E \ vectorspace (U + V)"
by (rule subspace.vectorspace) (rule sum_subspace)
subsection \<open>Direct sums\<close>
text \<open>
The sum of \<open>U\<close> and \<open>V\<close> is called \<^emph>\<open>direct\<close>, iff the zero element is the only
common element of \<open>U\<close> and \<open>V\<close>. For every element \<open>x\<close> of the direct sum of
\<open>U\<close> and \<open>V\<close> the decomposition in \<open>x = u + v\<close> with \<open>u \<in> U\<close> and \<open>v \<in> V\<close> is
unique.
\<close>
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 \<open>subspace U E\<close> \<open>vectorspace E\<close> by (rule subspace.vectorspace)
have V: "vectorspace V"
using \<open>subspace V E\<close> \<open>vectorspace E\<close> by (rule subspace.vectorspace)
from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
by (simp add: add_diff_swap)
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"
proof (rule add_minus_eq)
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"
proof (rule add_minus_eq [symmetric])
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 \<open>
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
\<open>y + a \<cdot> x\<^sub>0\<close> of the direct sum of a vectorspace \<open>H\<close> and the linear closure
of \<open>x\<^sub>0\<close> the components \<open>y \<in> H\<close> and \<open>a\<close> are uniquely determined.
\<close>
lemma decomp_H':
assumes "vectorspace E" "subspace H E"
assumes y1: "y1 \ H" and y2: "y2 \ H"
and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 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' \<in> H" by (simp add: mult_assoc2)
with \<open>x' \<notin> H\<close> 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 \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, 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 \<open>
Since for any element \<open>y + a \<cdot> x'\<close> of the direct sum of a vectorspace \<open>H\<close>
and the linear closure of \<open>x'\<close> the components \<open>y \<in> H\<close> and \<open>a\<close> are unique, it
follows from \<open>y \<in> H\<close> that \<open>a = 0\<close>.
\<close>
lemma decomp_H'_H:
assumes "vectorspace E" "subspace H E"
assumes t: "t \ H"
and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 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 \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, rule t, (rule x')+)
with t x' show "(y, a) = (y + a \ x', 0)" by simp
qed
qed
text \<open>
The components \<open>y \<in> H\<close> and \<open>a\<close> in \<open>y + a \<cdot> x'\<close> are unique, so the function
\<open>h'\<close> defined by \<open>h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>\<close> is definite.
\<close>
lemma h'_definite:
fixes H
assumes h'_def:
"\x. h' x =
(let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> 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' \<notin> H" "x' \<in> E" "x' \<noteq> 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 \<open>vectorspace E\<close>, rule \<open>subspace H E\<close>, (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
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