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src/HOL/Hahn_Banach/Normed_Space.thy

author | wenzelm |

Sun, 11 Sep 2011 22:55:26 +0200 | |

changeset 44887 | 7ca82df6e951 |

parent 31795 | be3e1cc5005c |

child 46867 | 0883804b67bb |

permissions | -rw-r--r-- |

misc tuning and clarification;

(* Title: HOL/Hahn_Banach/Normed_Space.thy Author: Gertrud Bauer, TU Munich *) header {* Normed vector spaces *} theory Normed_Space imports Subspace begin subsection {* Quasinorms *} text {* A \emph{seminorm} @{text "\<parallel>\<cdot>\<parallel>"} is a function on a real vector space into the reals that has the following properties: it is positive definite, absolute homogenous and subadditive. *} locale norm_syntax = fixes norm :: "'a \<Rightarrow> real" ("\<parallel>_\<parallel>") locale seminorm = var_V + norm_syntax + constrains V :: "'a\<Colon>{minus, plus, zero, uminus} set" assumes ge_zero [iff?]: "x \<in> V \<Longrightarrow> 0 \<le> \<parallel>x\<parallel>" and abs_homogenous [iff?]: "x \<in> V \<Longrightarrow> \<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" and subadditive [iff?]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" declare seminorm.intro [intro?] lemma (in seminorm) diff_subadditive: assumes "vectorspace V" shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x - y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" proof - interpret vectorspace V by fact assume x: "x \<in> V" and y: "y \<in> V" then have "x - y = x + - 1 \<cdot> y" by (simp add: diff_eq2 negate_eq2a) also from x y have "\<parallel>\<dots>\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>- 1 \<cdot> y\<parallel>" by (simp add: subadditive) also from y have "\<parallel>- 1 \<cdot> y\<parallel> = \<bar>- 1\<bar> * \<parallel>y\<parallel>" by (rule abs_homogenous) also have "\<dots> = \<parallel>y\<parallel>" by simp finally show ?thesis . qed lemma (in seminorm) minus: assumes "vectorspace V" shows "x \<in> V \<Longrightarrow> \<parallel>- x\<parallel> = \<parallel>x\<parallel>" proof - interpret vectorspace V by fact assume x: "x \<in> V" then have "- x = - 1 \<cdot> x" by (simp only: negate_eq1) also from x have "\<parallel>\<dots>\<parallel> = \<bar>- 1\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous) also have "\<dots> = \<parallel>x\<parallel>" by simp finally show ?thesis . qed subsection {* Norms *} text {* A \emph{norm} @{text "\<parallel>\<cdot>\<parallel>"} is a seminorm that maps only the @{text 0} vector to @{text 0}. *} locale norm = seminorm + assumes zero_iff [iff]: "x \<in> V \<Longrightarrow> (\<parallel>x\<parallel> = 0) = (x = 0)" subsection {* Normed vector spaces *} text {* A vector space together with a norm is called a \emph{normed space}. *} locale normed_vectorspace = vectorspace + norm declare normed_vectorspace.intro [intro?] lemma (in normed_vectorspace) gt_zero [intro?]: assumes x: "x \<in> V" and neq: "x \<noteq> 0" shows "0 < \<parallel>x\<parallel>" proof - from x have "0 \<le> \<parallel>x\<parallel>" .. also have "0 \<noteq> \<parallel>x\<parallel>" proof assume "0 = \<parallel>x\<parallel>" with x have "x = 0" by simp with neq show False by contradiction qed finally show ?thesis . qed text {* Any subspace of a normed vector space is again a normed vectorspace. *} lemma subspace_normed_vs [intro?]: fixes F E norm assumes "subspace F E" "normed_vectorspace E norm" shows "normed_vectorspace F norm" proof - interpret subspace F E by fact interpret normed_vectorspace E norm by fact show ?thesis proof show "vectorspace F" by (rule vectorspace) unfold_locales next have "Normed_Space.norm E norm" .. with subset show "Normed_Space.norm F norm" by (simp add: norm_def seminorm_def norm_axioms_def) qed qed end