src/HOL/Real/HahnBanach/Linearform.thy
author wenzelm
Thu Aug 22 20:49:43 2002 +0200 (2002-08-22)
changeset 13515 a6a7025fd7e8
parent 12018 ec054019c910
child 13547 bf399f3bd7dc
permissions -rw-r--r--
updated to use locales (still some rough edges);
     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 locale (open) vectorspace_linearform =
    20   vectorspace + linearform
    21 
    22 lemma (in vectorspace_linearform) neg [iff]:
    23   "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 vectorspace_linearform) diff [iff]:
    33   "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
    34 proof -
    35   assume x: "x \<in> V" and y: "y \<in> V"
    36   hence "x - y = x + - y" by (rule diff_eq1)
    37   also have "f ... = f x + f (- y)"
    38     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 vectorspace_linearform) linearform_zero [iff]:
    46   "f 0 = 0"
    47 proof -
    48   have "f 0 = f (0 - 0)" by simp
    49   also have "\<dots> = f 0 - f 0" by (rule diff) simp_all
    50   also have "\<dots> = 0" by simp
    51   finally show ?thesis .
    52 qed
    53 
    54 end