src/HOL/Real/HahnBanach/Linearform.thy
 author wenzelm Thu Aug 29 16:08:30 2002 +0200 (2002-08-29) changeset 13547 bf399f3bd7dc parent 13515 a6a7025fd7e8 child 14254 342634f38451 permissions -rw-r--r--
tuned;
     1 (*  Title:      HOL/Real/HahnBanach/Linearform.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer, TU Munich

     4 *)

     5

     6 header {* Linearforms *}

     7

     8 theory Linearform = VectorSpace:

     9

    10 text {*

    11   A \emph{linear form} is a function on a vector space into the reals

    12   that is additive and multiplicative.

    13 *}

    14

    15 locale linearform = var V + var f +

    16   assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"

    17     and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"

    18

    19 lemma (in linearform) neg [iff]:

    20   includes vectorspace

    21   shows "x \<in> V \<Longrightarrow> f (- x) = - f x"

    22 proof -

    23   assume x: "x \<in> V"

    24   hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)

    25   also from x have "... = (- 1) * (f x)" by (rule mult)

    26   also from x have "... = - (f x)" by simp

    27   finally show ?thesis .

    28 qed

    29

    30 lemma (in linearform) diff [iff]:

    31   includes vectorspace

    32   shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"

    33 proof -

    34   assume x: "x \<in> V" and y: "y \<in> V"

    35   hence "x - y = x + - y" by (rule diff_eq1)

    36   also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)

    37   also from _ y have "f (- y) = - f y" by (rule neg)

    38   finally show ?thesis by simp

    39 qed

    40

    41 text {* Every linear form yields @{text 0} for the @{text 0} vector. *}

    42

    43 lemma (in linearform) zero [iff]:

    44   includes vectorspace

    45   shows "f 0 = 0"

    46 proof -

    47   have "f 0 = f (0 - 0)" by simp

    48   also have "\<dots> = f 0 - f 0" by (rule diff) simp_all

    49   also have "\<dots> = 0" by simp

    50   finally show ?thesis .

    51 qed

    52

    53 end