(* Title: HOL/Hahn_Banach/Normed_Space.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Normed vector spaces\<close>
theory Normed_Space
imports Subspace
begin
subsection \<open>Quasinorms\<close>
text \<open>
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 homogeneous and subadditive.
\<close>
locale seminorm =
fixes V :: "'a\<Colon>{minus, plus, zero, uminus} set"
fixes norm :: "'a \<Rightarrow> real" ("\<parallel>_\<parallel>")
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 \<open>Norms\<close>
text \<open>
A \emph{norm} @{text "\<parallel>\<cdot>\<parallel>"} is a seminorm that maps only the
@{text 0} vector to @{text 0}.
\<close>
locale norm = seminorm +
assumes zero_iff [iff]: "x \<in> V \<Longrightarrow> (\<parallel>x\<parallel> = 0) = (x = 0)"
subsection \<open>Normed vector spaces\<close>
text \<open>
A vector space together with a norm is called a \emph{normed
space}.
\<close>
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 \<open>
Any subspace of a normed vector space is again a normed vectorspace.
\<close>
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