(* Title: HOL/Real/HahnBanach/Linearform.thy
ID: $Id$
Author: Gertrud Bauer, TU Munich
*)
header {* Linearforms *}
theory Linearform imports VectorSpace begin
text {*
A \emph{linear form} is a function on a vector space into the reals
that is additive and multiplicative.
*}
locale linearform = var V + var 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]:
includes vectorspace
shows "x \ V \ f (- x) = - f x"
proof -
assume x: "x \ V"
hence "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]:
includes vectorspace
shows "x \ V \ y \ V \ f (x - y) = f x - f y"
proof -
assume x: "x \ V" and y: "y \ V"
hence "x - y = x + - y" by (rule diff_eq1)
also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)
also from _ y have "f (- y) = - f y" by (rule neg)
finally show ?thesis by simp
qed
text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
lemma (in linearform) zero [iff]:
includes vectorspace
shows "f 0 = 0"
proof -
have "f 0 = f (0 - 0)" by simp
also have "\ = f 0 - f 0" by (rule diff) simp_all
also have "\ = 0" by simp
finally show ?thesis .
qed
end