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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 *)

6 header {* Linearforms *}

8 theory Linearform = VectorSpace:

10 text {*

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

12 that is additive and multiplicative.

13 *}

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"

19 locale (open) vectorspace_linearform =

20 vectorspace + linearform

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

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

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

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

54 end