src/HOL/Real/HahnBanach/Linearform.thy
 author haftmann Fri Jun 17 16:12:49 2005 +0200 (2005-06-17) changeset 16417 9bc16273c2d4 parent 14254 342634f38451 child 23378 1d138d6bb461 permissions -rw-r--r--
migrated theory headers to new format
     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 imports VectorSpace begin

     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 declare linearform.intro [intro?]

    20

    21 lemma (in linearform) neg [iff]:

    22   includes vectorspace

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

    24 proof -

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

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

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

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

    29   finally show ?thesis .

    30 qed

    31

    32 lemma (in linearform) diff [iff]:

    33   includes vectorspace

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

    35 proof -

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

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

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

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

    40   finally show ?thesis by simp

    41 qed

    42

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

    44

    45 lemma (in linearform) zero [iff]:

    46   includes vectorspace

    47   shows "f 0 = 0"

    48 proof -

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

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

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

    52   finally show ?thesis .

    53 qed

    54

    55 end