(* 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" thenhave"f (- x) = f ((- 1) \ x)" by (simp add: negate_eq1) alsofrom x have"\ = (- 1) * (f x)" by (rule mult) alsofrom x have"\ = - (f x)" by simp finallyshow ?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" thenhave"x - y = x + - y"by (rule diff_eq1) alsohave"f \ = f x + f (- y)" by (rule add) (simp_all add: x y) alsohave"f (- y) = - f y"using\<open>vectorspace V\<close> y by (rule neg) finallyshow ?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 alsohave"\ = f 0 - f 0" using \vectorspace V\ by (rule diff) simp_all alsohave"\ = 0" by simp finallyshow ?thesis . qed
end
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