src/HOL/Hahn_Banach/Normed_Space.thy
 author wenzelm Sun Sep 11 22:55:26 2011 +0200 (2011-09-11) changeset 44887 7ca82df6e951 parent 31795 be3e1cc5005c child 46867 0883804b67bb permissions -rw-r--r--
misc tuning and clarification;
     1 (*  Title:      HOL/Hahn_Banach/Normed_Space.thy

     2     Author:     Gertrud Bauer, TU Munich

     3 *)

     4

     5 header {* Normed vector spaces *}

     6

     7 theory Normed_Space

     8 imports Subspace

     9 begin

    10

    11 subsection {* Quasinorms *}

    12

    13 text {*

    14   A \emph{seminorm} @{text "\<parallel>\<cdot>\<parallel>"} is a function on a real vector space

    15   into the reals that has the following properties: it is positive

    16   definite, absolute homogenous and subadditive.

    17 *}

    18

    19 locale norm_syntax =

    20   fixes norm :: "'a \<Rightarrow> real"    ("\<parallel>_\<parallel>")

    21

    22 locale seminorm = var_V + norm_syntax +

    23   constrains V :: "'a\<Colon>{minus, plus, zero, uminus} set"

    24   assumes ge_zero [iff?]: "x \<in> V \<Longrightarrow> 0 \<le> \<parallel>x\<parallel>"

    25     and abs_homogenous [iff?]: "x \<in> V \<Longrightarrow> \<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"

    26     and subadditive [iff?]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"

    27

    28 declare seminorm.intro [intro?]

    29

    30 lemma (in seminorm) diff_subadditive:

    31   assumes "vectorspace V"

    32   shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x - y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"

    33 proof -

    34   interpret vectorspace V by fact

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

    36   then have "x - y = x + - 1 \<cdot> y"

    37     by (simp add: diff_eq2 negate_eq2a)

    38   also from x y have "\<parallel>\<dots>\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>- 1 \<cdot> y\<parallel>"

    39     by (simp add: subadditive)

    40   also from y have "\<parallel>- 1 \<cdot> y\<parallel> = \<bar>- 1\<bar> * \<parallel>y\<parallel>"

    41     by (rule abs_homogenous)

    42   also have "\<dots> = \<parallel>y\<parallel>" by simp

    43   finally show ?thesis .

    44 qed

    45

    46 lemma (in seminorm) minus:

    47   assumes "vectorspace V"

    48   shows "x \<in> V \<Longrightarrow> \<parallel>- x\<parallel> = \<parallel>x\<parallel>"

    49 proof -

    50   interpret vectorspace V by fact

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

    52   then have "- x = - 1 \<cdot> x" by (simp only: negate_eq1)

    53   also from x have "\<parallel>\<dots>\<parallel> = \<bar>- 1\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)

    54   also have "\<dots> = \<parallel>x\<parallel>" by simp

    55   finally show ?thesis .

    56 qed

    57

    58

    59 subsection {* Norms *}

    60

    61 text {*

    62   A \emph{norm} @{text "\<parallel>\<cdot>\<parallel>"} is a seminorm that maps only the

    63   @{text 0} vector to @{text 0}.

    64 *}

    65

    66 locale norm = seminorm +

    67   assumes zero_iff [iff]: "x \<in> V \<Longrightarrow> (\<parallel>x\<parallel> = 0) = (x = 0)"

    68

    69

    70 subsection {* Normed vector spaces *}

    71

    72 text {*

    73   A vector space together with a norm is called a \emph{normed

    74   space}.

    75 *}

    76

    77 locale normed_vectorspace = vectorspace + norm

    78

    79 declare normed_vectorspace.intro [intro?]

    80

    81 lemma (in normed_vectorspace) gt_zero [intro?]:

    82   assumes x: "x \<in> V" and neq: "x \<noteq> 0"

    83   shows "0 < \<parallel>x\<parallel>"

    84 proof -

    85   from x have "0 \<le> \<parallel>x\<parallel>" ..

    86   also have "0 \<noteq> \<parallel>x\<parallel>"

    87   proof

    88     assume "0 = \<parallel>x\<parallel>"

    89     with x have "x = 0" by simp

    90     with neq show False by contradiction

    91   qed

    92   finally show ?thesis .

    93 qed

    94

    95 text {*

    96   Any subspace of a normed vector space is again a normed vectorspace.

    97 *}

    98

    99 lemma subspace_normed_vs [intro?]:

   100   fixes F E norm

   101   assumes "subspace F E" "normed_vectorspace E norm"

   102   shows "normed_vectorspace F norm"

   103 proof -

   104   interpret subspace F E by fact

   105   interpret normed_vectorspace E norm by fact

   106   show ?thesis

   107   proof

   108     show "vectorspace F" by (rule vectorspace) unfold_locales

   109   next

   110     have "Normed_Space.norm E norm" ..

   111     with subset show "Normed_Space.norm F norm"

   112       by (simp add: norm_def seminorm_def norm_axioms_def)

   113   qed

   114 qed

   115

   116 end