(* Title: HOL/Hahn_Banach/Function_Norm.thy
Author: Gertrud Bauer, TU Munich
*)
header {* The norm of a function *}
theory Function_Norm
imports Normed_Space Function_Order
begin
subsection {* Continuous linear forms*}
text {*
A linear form @{text f} on a normed vector space @{text "(V, \<parallel>\<cdot>\<parallel>)"}
is \emph{continuous}, iff it is bounded, i.e.
\begin{center}
@{text "\<exists>c \<in> R. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
In our application no other functions than linear forms are
considered, so we can define continuous linear forms as bounded
linear forms:
*}
locale continuous = linearform +
fixes norm :: "_ \<Rightarrow> real" ("\<parallel>_\<parallel>")
assumes bounded: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
declare continuous.intro [intro?] continuous_axioms.intro [intro?]
lemma continuousI [intro]:
fixes norm :: "_ \<Rightarrow> real" ("\<parallel>_\<parallel>")
assumes "linearform V f"
assumes r: "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
shows "continuous V f norm"
proof
show "linearform V f" by fact
from r have "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" by blast
then show "continuous_axioms V f norm" ..
qed
subsection {* The norm of a linear form *}
text {*
The least real number @{text c} for which holds
\begin{center}
@{text "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
is called the \emph{norm} of @{text f}.
For non-trivial vector spaces @{text "V \<noteq> {0}"} the norm can be
defined as
\begin{center}
@{text "\<parallel>f\<parallel> = \<sup>x \<noteq> 0. \<bar>f x\<bar> / \<parallel>x\<parallel>"}
\end{center}
For the case @{text "V = {0}"} the supremum would be taken from an
empty set. Since @{text \<real>} is unbounded, there would be no supremum.
To avoid this situation it must be guaranteed that there is an
element in this set. This element must be @{text "{} \<ge> 0"} so that
@{text fn_norm} has the norm properties. Furthermore it does not
have to change the norm in all other cases, so it must be @{text 0},
as all other elements are @{text "{} \<ge> 0"}.
Thus we define the set @{text B} where the supremum is taken from as
follows:
\begin{center}
@{text "{0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel>. x \<noteq> 0 \<and> x \<in> F}"}
\end{center}
@{text fn_norm} is equal to the supremum of @{text B}, if the
supremum exists (otherwise it is undefined).
*}
locale fn_norm =
fixes norm :: "_ \<Rightarrow> real" ("\<parallel>_\<parallel>")
fixes B defines "B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"
fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
defines "\<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
locale normed_vectorspace_with_fn_norm = normed_vectorspace + fn_norm
lemma (in fn_norm) B_not_empty [intro]: "0 \<in> B V f"
by (simp add: B_def)
text {*
The following lemma states that every continuous linear form on a
normed space @{text "(V, \<parallel>\<cdot>\<parallel>)"} has a function norm.
*}
lemma (in normed_vectorspace_with_fn_norm) fn_norm_works:
assumes "continuous V f norm"
shows "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
proof -
interpret continuous V f norm by fact
txt {* The existence of the supremum is shown using the
completeness of the reals. Completeness means, that every
non-empty bounded set of reals has a supremum. *}
have "\<exists>a. lub (B V f) a"
proof (rule real_complete)
txt {* First we have to show that @{text B} is non-empty: *}
have "0 \<in> B V f" ..
then show "\<exists>x. x \<in> B V f" ..
txt {* Then we have to show that @{text B} is bounded: *}
show "\<exists>c. \<forall>y \<in> B V f. y \<le> c"
proof -
txt {* We know that @{text f} is bounded by some value @{text c}. *}
from bounded obtain c where c: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
txt {* To prove the thesis, we have to show that there is some
@{text b}, such that @{text "y \<le> b"} for all @{text "y \<in>
B"}. Due to the definition of @{text B} there are two cases. *}
def b \<equiv> "max c 0"
have "\<forall>y \<in> B V f. y \<le> b"
proof
fix y assume y: "y \<in> B V f"
show "y \<le> b"
proof cases
assume "y = 0"
then show ?thesis unfolding b_def by arith
next
txt {* The second case is @{text "y = \<bar>f x\<bar> / \<parallel>x\<parallel>"} for some
@{text "x \<in> V"} with @{text "x \<noteq> 0"}. *}
assume "y \<noteq> 0"
with y obtain x where y_rep: "y = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
and x: "x \<in> V" and neq: "x \<noteq> 0"
by (auto simp add: B_def divide_inverse)
from x neq have gt: "0 < \<parallel>x\<parallel>" ..
txt {* The thesis follows by a short calculation using the
fact that @{text f} is bounded. *}
note y_rep
also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
proof (rule mult_right_mono)
from c x show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
from gt have "0 < inverse \<parallel>x\<parallel>"
by (rule positive_imp_inverse_positive)
then show "0 \<le> inverse \<parallel>x\<parallel>" by (rule order_less_imp_le)
qed
also have "\<dots> = c * (\<parallel>x\<parallel> * inverse \<parallel>x\<parallel>)"
by (rule Groups.mult_assoc)
also
from gt have "\<parallel>x\<parallel> \<noteq> 0" by simp
then have "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by simp
also have "c * 1 \<le> b" by (simp add: b_def)
finally show "y \<le> b" .
qed
qed
then show ?thesis ..
qed
qed
then show ?thesis unfolding fn_norm_def by (rule the_lubI_ex)
qed
lemma (in normed_vectorspace_with_fn_norm) fn_norm_ub [iff?]:
assumes "continuous V f norm"
assumes b: "b \<in> B V f"
shows "b \<le> \<parallel>f\<parallel>\<hyphen>V"
proof -
interpret continuous V f norm by fact
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
using `continuous V f norm` by (rule fn_norm_works)
from this and b show ?thesis ..
qed
lemma (in normed_vectorspace_with_fn_norm) fn_norm_leastB:
assumes "continuous V f norm"
assumes b: "\<And>b. b \<in> B V f \<Longrightarrow> b \<le> y"
shows "\<parallel>f\<parallel>\<hyphen>V \<le> y"
proof -
interpret continuous V f norm by fact
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
using `continuous V f norm` by (rule fn_norm_works)
from this and b show ?thesis ..
qed
text {* The norm of a continuous function is always @{text "\<ge> 0"}. *}
lemma (in normed_vectorspace_with_fn_norm) fn_norm_ge_zero [iff]:
assumes "continuous V f norm"
shows "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
proof -
interpret continuous V f norm by fact
txt {* The function norm is defined as the supremum of @{text B}.
So it is @{text "\<ge> 0"} if all elements in @{text B} are @{text "\<ge>
0"}, provided the supremum exists and @{text B} is not empty. *}
have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
using `continuous V f norm` by (rule fn_norm_works)
moreover have "0 \<in> B V f" ..
ultimately show ?thesis ..
qed
text {*
\medskip The fundamental property of function norms is:
\begin{center}
@{text "\<bar>f x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
\end{center}
*}
lemma (in normed_vectorspace_with_fn_norm) fn_norm_le_cong:
assumes "continuous V f norm" "linearform V f"
assumes x: "x \<in> V"
shows "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
proof -
interpret continuous V f norm by fact
interpret linearform V f by fact
show ?thesis
proof cases
assume "x = 0"
then have "\<bar>f x\<bar> = \<bar>f 0\<bar>" by simp
also have "f 0 = 0" by rule unfold_locales
also have "\<bar>\<dots>\<bar> = 0" by simp
also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
using `continuous V f norm` by (rule fn_norm_ge_zero)
from x have "0 \<le> norm x" ..
with a have "0 \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" by (simp add: zero_le_mult_iff)
finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" .
next
assume "x \<noteq> 0"
with x have neq: "\<parallel>x\<parallel> \<noteq> 0" by simp
then have "\<bar>f x\<bar> = (\<bar>f x\<bar> * inverse \<parallel>x\<parallel>) * \<parallel>x\<parallel>" by simp
also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
proof (rule mult_right_mono)
from x show "0 \<le> \<parallel>x\<parallel>" ..
from x and neq have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<in> B V f"
by (auto simp add: B_def divide_inverse)
with `continuous V f norm` show "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>V"
by (rule fn_norm_ub)
qed
finally show ?thesis .
qed
qed
text {*
\medskip The function norm is the least positive real number for
which the following inequation holds:
\begin{center}
@{text "\<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
*}
lemma (in normed_vectorspace_with_fn_norm) fn_norm_least [intro?]:
assumes "continuous V f norm"
assumes ineq: "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" and ge: "0 \<le> c"
shows "\<parallel>f\<parallel>\<hyphen>V \<le> c"
proof -
interpret continuous V f norm by fact
show ?thesis
proof (rule fn_norm_leastB [folded B_def fn_norm_def])
fix b assume b: "b \<in> B V f"
show "b \<le> c"
proof cases
assume "b = 0"
with ge show ?thesis by simp
next
assume "b \<noteq> 0"
with b obtain x where b_rep: "b = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
and x_neq: "x \<noteq> 0" and x: "x \<in> V"
by (auto simp add: B_def divide_inverse)
note b_rep
also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
proof (rule mult_right_mono)
have "0 < \<parallel>x\<parallel>" using x x_neq ..
then show "0 \<le> inverse \<parallel>x\<parallel>" by simp
from x show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" by (rule ineq)
qed
also have "\<dots> = c"
proof -
from x_neq and x have "\<parallel>x\<parallel> \<noteq> 0" by simp
then show ?thesis by simp
qed
finally show ?thesis .
qed
qed (insert `continuous V f norm`, simp_all add: continuous_def)
qed
end