(*  Title:      HOL/Real/HahnBanach/NormedSpace.thy

    ID:         $Id$

    Author:     Gertrud Bauer, TU Munich

*)

header {* Normed vector spaces *}

theory NormedSpace =  Subspace:

syntax 

  abs :: "real \<Rightarrow> real" ("|_|")

subsection {* Quasinorms *}

text{* A \emph{seminorm} $\norm{\cdot}$ is a function on a real vector

space into the reals that has the following properties: It is positive

definite, absolute homogenous and subadditive.  *}

constdefs

  is_seminorm :: "['a::{plus, minus, zero} set, 'a => real] => bool"

  "is_seminorm V norm == \<forall>x \<in>  V. \<forall>y \<in> V. \<forall>a. 

        #0 <= norm x 

      \<and> norm (a \<cdot> x) = |a| * norm x

      \<and> norm (x + y) <= norm x + norm y"

lemma is_seminormI [intro]: 

  "[| !! x y a. [| x \<in> V; y \<in> V|] ==> #0 <= norm x;

  !! x a. x \<in> V ==> norm (a \<cdot> x) = |a| * norm x;

  !! x y. [|x \<in> V; y \<in> V |] ==> norm (x + y) <= norm x + norm y |] 

  ==> is_seminorm V norm"

  by (unfold is_seminorm_def, force)

lemma seminorm_ge_zero [intro?]:

  "[| is_seminorm V norm; x \<in> V |] ==> #0 <= norm x"

  by (unfold is_seminorm_def, force)

lemma seminorm_abs_homogenous: 

  "[| is_seminorm V norm; x \<in> V |] 

  ==> norm (a \<cdot> x) = |a| * norm x"

  by (unfold is_seminorm_def, force)

lemma seminorm_subadditive: 

  "[| is_seminorm V norm; x \<in> V; y \<in> V |] 

  ==> norm (x + y) <= norm x + norm y"

  by (unfold is_seminorm_def, force)

lemma seminorm_diff_subadditive: 

  "[| is_seminorm V norm; x \<in> V; y \<in> V; is_vectorspace V |] 

  ==> norm (x - y) <= norm x + norm y"

proof -

  assume "is_seminorm V norm" "x \<in> V" "y \<in> V" "is_vectorspace V"

  have "norm (x - y) = norm (x + - #1 \<cdot> y)"  

    by (simp! add: diff_eq2 negate_eq2a)

  also have "... <= norm x + norm  (- #1 \<cdot> y)" 

    by (simp! add: seminorm_subadditive)

  also have "norm (- #1 \<cdot> y) = |- #1| * norm y" 

    by (rule seminorm_abs_homogenous)

  also have "|- #1| = (#1::real)" by (rule abs_minus_one)

  finally show "norm (x - y) <= norm x + norm y" by simp

qed

lemma seminorm_minus: 

  "[| is_seminorm V norm; x \<in> V; is_vectorspace V |] 

  ==> norm (- x) = norm x"

proof -

  assume "is_seminorm V norm" "x \<in> V" "is_vectorspace V"

  have "norm (- x) = norm (- #1 \<cdot> x)" by (simp! only: negate_eq1)

  also have "... = |- #1| * norm x" 

    by (rule seminorm_abs_homogenous)

  also have "|- #1| = (#1::real)" by (rule abs_minus_one)

  finally show "norm (- x) = norm x" by simp

qed

subsection {* Norms *}

text{* A \emph{norm} $\norm{\cdot}$ is a seminorm that maps only the

$\zero$ vector to $0$. *}

constdefs

  is_norm :: "['a::{plus, minus, zero} set, 'a => real] => bool"

  "is_norm V norm == \<forall>x \<in> V.  is_seminorm V norm 

      \<and> (norm x = #0) = (x = 0)"

lemma is_normI [intro]: 

  "\<forall>x \<in> V.  is_seminorm V norm  \<and> (norm x = #0) = (x = 0) 

  ==> is_norm V norm" by (simp only: is_norm_def)

lemma norm_is_seminorm [intro?]: 

  "[| is_norm V norm; x \<in> V |] ==> is_seminorm V norm"

  by (unfold is_norm_def, force)

lemma norm_zero_iff: 

  "[| is_norm V norm; x \<in> V |] ==> (norm x = #0) = (x = 0)"

  by (unfold is_norm_def, force)

lemma norm_ge_zero [intro?]:

  "[|is_norm V norm; x \<in> V |] ==> #0 <= norm x"

  by (unfold is_norm_def is_seminorm_def, force)

subsection {* Normed vector spaces *}

text{* A vector space together with a norm is called

a \emph{normed space}. *}

constdefs

  is_normed_vectorspace :: 

  "['a::{plus, minus, zero} set, 'a => real] => bool"

  "is_normed_vectorspace V norm ==

      is_vectorspace V \<and> is_norm V norm"

lemma normed_vsI [intro]: 

  "[| is_vectorspace V; is_norm V norm |] 

  ==> is_normed_vectorspace V norm"

  by (unfold is_normed_vectorspace_def) blast 

lemma normed_vs_vs [intro?]: 

  "is_normed_vectorspace V norm ==> is_vectorspace V"

  by (unfold is_normed_vectorspace_def) force

lemma normed_vs_norm [intro?]: 

  "is_normed_vectorspace V norm ==> is_norm V norm"

  by (unfold is_normed_vectorspace_def, elim conjE)

lemma normed_vs_norm_ge_zero [intro?]: 

  "[| is_normed_vectorspace V norm; x \<in> V |] ==> #0 <= norm x"

  by (unfold is_normed_vectorspace_def, rule, elim conjE)

lemma normed_vs_norm_gt_zero [intro?]: 

  "[| is_normed_vectorspace V norm; x \<in> V; x \<noteq> 0 |] ==> #0 < norm x"

proof (unfold is_normed_vectorspace_def, elim conjE)

  assume "x \<in> V" "x \<noteq> 0" "is_vectorspace V" "is_norm V norm"

  have "#0 <= norm x" ..

  also have "#0 \<noteq> norm x"

  proof

    presume "norm x = #0"

    also have "?this = (x = 0)" by (rule norm_zero_iff)

    finally have "x = 0" .

    thus "False" by contradiction

  qed (rule sym)

  finally show "#0 < norm x" .

qed

lemma normed_vs_norm_abs_homogenous [intro?]: 

  "[| is_normed_vectorspace V norm; x \<in> V |] 

  ==> norm (a \<cdot> x) = |a| * norm x"

  by (rule seminorm_abs_homogenous, rule norm_is_seminorm, 

      rule normed_vs_norm)

lemma normed_vs_norm_subadditive [intro?]: 

  "[| is_normed_vectorspace V norm; x \<in> V; y \<in> V |] 

  ==> norm (x + y) <= norm x + norm y"

  by (rule seminorm_subadditive, rule norm_is_seminorm, 

     rule normed_vs_norm)

text{* Any subspace of a normed vector space is again a 

normed vectorspace.*}

lemma subspace_normed_vs [intro?]: 

  "[| is_vectorspace E; is_subspace F E;

  is_normed_vectorspace E norm |] ==> is_normed_vectorspace F norm"

proof (rule normed_vsI)

  assume "is_subspace F E" "is_vectorspace E" 

         "is_normed_vectorspace E norm"

  show "is_vectorspace F" ..

  show "is_norm F norm" 

  proof (intro is_normI ballI conjI)

    show "is_seminorm F norm" 

    proof

      fix x y a presume "x \<in> E"

      show "#0 <= norm x" ..

      show "norm (a \<cdot> x) = |a| * norm x" ..

      presume "y \<in> E"

      show "norm (x + y) <= norm x + norm y" ..

    qed (simp!)+

    fix x assume "x \<in> F"

    show "(norm x = #0) = (x = 0)" 

    proof (rule norm_zero_iff)

      show "is_norm E norm" ..

    qed (simp!)

  qed

qed

end