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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 *)

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 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

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

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

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

53 end