(* Title: Library/Operator_Norm.thy
Author: Amine Chaieb, University of Cambridge
*)
header {* Operator Norm *}
theory Operator_Norm
imports Euclidean_Space
begin
definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
lemma norm_bound_generalize:
fixes f:: "real ^'n \<Rightarrow> real^'m"
assumes lf: "linear f"
shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume H: ?rhs
{fix x :: "real^'n" assume x: "norm x = 1"
from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
then have ?lhs by blast }
moreover
{assume H: ?lhs
from H[rule_format, of "basis arbitrary"]
have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
{fix x :: "real ^'n"
{assume "x = 0"
then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
moreover
{assume x0: "x \<noteq> 0"
hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
let ?c = "1/ norm x"
have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0)
with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast
hence "?c * norm (f x) \<le> b"
by (simp add: linear_cmul[OF lf])
hence "norm (f x) \<le> b * norm x"
using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
ultimately have "norm (f x) \<le> b * norm x" by blast}
then have ?rhs by blast}
ultimately show ?thesis by blast
qed
lemma onorm:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "norm (f x) <= onorm f * norm x"
and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
proof-
{
let ?S = "{norm (f x) |x. norm x = 1}"
have Se: "?S \<noteq> {}" using norm_basis by auto
from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
{from Sup[OF Se b, unfolded onorm_def[symmetric]]
show "norm (f x) <= onorm f * norm x"
apply -
apply (rule spec[where x = x])
unfolding norm_bound_generalize[OF lf, symmetric]
by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
{
show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
using Sup[OF Se b, unfolded onorm_def[symmetric]]
unfolding norm_bound_generalize[OF lf, symmetric]
by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
}
qed
lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
using onorm[OF lf]
apply (auto simp add: onorm_pos_le)
apply atomize
apply (erule allE[where x="0::real"])
using onorm_pos_le[OF lf]
apply arith
done
lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^'m)) = norm y"
proof-
let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
by(auto intro: vector_choose_size set_ext)
show ?thesis
unfolding onorm_def th
apply (rule Sup_unique) by (simp_all add: setle_def)
qed
lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
unfolding onorm_eq_0[OF lf, symmetric]
using onorm_pos_le[OF lf] by arith
lemma onorm_compose:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
and lg: "linear (g::real^'k \<Rightarrow> real^'n)"
shows "onorm (f o g) <= onorm f * onorm g"
apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
unfolding o_def
apply (subst mult_assoc)
apply (rule order_trans)
apply (rule onorm(1)[OF lf])
apply (rule mult_mono1)
apply (rule onorm(1)[OF lg])
apply (rule onorm_pos_le[OF lf])
done
lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
shows "onorm (\<lambda>x. - f x) \<le> onorm f"
using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
unfolding norm_minus_cancel by metis
lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
shows "onorm (\<lambda>x. - f x) = onorm f"
using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
by simp
lemma onorm_triangle:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
apply (rule order_trans)
apply (rule norm_triangle_ineq)
apply (simp add: distrib)
apply (rule add_mono)
apply (rule onorm(1)[OF lf])
apply (rule onorm(1)[OF lg])
done
lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
\<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
apply (rule order_trans)
apply (rule onorm_triangle)
apply assumption+
done
lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
==> onorm(\<lambda>x. f x + g x) < e"
apply (rule order_le_less_trans)
apply (rule onorm_triangle)
by assumption+
end