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(* Title: HOL/Hahn_Banach/Linearform.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Linearforms\<close>
theory Linearform
imports Vector_Space
begin
text \<open>
A \<^emph>\<open>linear form\<close> is a function on a vector space into the reals that is
additive and multiplicative.
\<close>
locale linearform =
fixes V :: "'a::{minus, plus, zero, uminus} set" and f
assumes add [iff]: "x \ V \ y \ V \ f (x + y) = f x + f y"
and mult [iff]: "x \ V \ f (a \ x) = a * f x"
declare linearform.intro [intro?]
lemma (in linearform) neg [iff]:
assumes "vectorspace V"
shows "x \ V \ f (- x) = - f x"
proof -
interpret vectorspace V by fact
assume x: "x \ V"
then have "f (- x) = f ((- 1) \ x)" by (simp add: negate_eq1)
also from x have "\ = (- 1) * (f x)" by (rule mult)
also from x have "\ = - (f x)" by simp
finally show ?thesis .
qed
lemma (in linearform) diff [iff]:
assumes "vectorspace V"
shows "x \ V \ y \ V \ f (x - y) = f x - f y"
proof -
interpret vectorspace V by fact
assume x: "x \ V" and y: "y \ V"
then have "x - y = x + - y" by (rule diff_eq1)
also have "f \ = f x + f (- y)" by (rule add) (simp_all add: x y)
also have "f (- y) = - f y" using \<open>vectorspace V\<close> y by (rule neg)
finally show ?thesis by simp
qed
text \<open>Every linear form yields \<open>0\<close> for the \<open>0\<close> vector.\<close>
lemma (in linearform) zero [iff]:
assumes "vectorspace V"
shows "f 0 = 0"
proof -
interpret vectorspace V by fact
have "f 0 = f (0 - 0)" by simp
also have "\ = f 0 - f 0" using \vectorspace V\ by (rule diff) simp_all
also have "\ = 0" by simp
finally show ?thesis .
qed
end
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