(* Title: HOL/Real/HahnBanach/FunctionNorm.thy
ID: $Id$
Author: Gertrud Bauer, TU Munich
*)
header {* The norm of a function *};
theory FunctionNorm = NormedSpace + FunctionOrder:;
subsection {* Continous linearforms*};
text{* A linearform $f$ on a normed vector space $(V, \norm{\cdot})$
is \emph{continous}, iff it is bounded, i.~e.
\[\exists\ap c\in R.\ap \forall\ap x\in V.\ap
|f\ap x| \leq c \cdot \norm x\]
In our application no other functions than linearforms are considered,
so we can define continous linearforms as follows:
*};
constdefs
is_continous ::
"['a::{minus, plus} set, 'a => real, 'a => real] => bool"
"is_continous V norm f ==
is_linearform V f & (EX c. ALL x:V. rabs (f x) <= c * norm x)";
lemma continousI [intro]:
"[| is_linearform V f; !! x. x:V ==> rabs (f x) <= c * norm x |]
==> is_continous V norm f";
proof (unfold is_continous_def, intro exI conjI ballI);
assume r: "!! x. x:V ==> rabs (f x) <= c * norm x";
fix x; assume "x:V"; show "rabs (f x) <= c * norm x"; by (rule r);
qed;
lemma continous_linearform [intro!!]:
"is_continous V norm f ==> is_linearform V f";
by (unfold is_continous_def) force;
lemma continous_bounded [intro!!]:
"is_continous V norm f
==> EX c. ALL x:V. rabs (f x) <= c * norm x";
by (unfold is_continous_def) force;
subsection{* The norm of a linearform *};
text{* The least real number $c$ for which holds
\[\forall\ap x\in V.\ap |f\ap x| \leq c \cdot \norm x\]
is called the \emph{norm} of $f$.
For the non-trivial vector space $V$ the norm can be defined as
\[\fnorm {f} =\sup_{x\neq\zero}\frac{|f\ap x|}{\norm x} \]
For the case that the vector space $V$ contains only the $\zero$
vector, the set $B$ this supremum is taken from would be empty, and
any real number is a supremum of $B$. So it must be guarateed that
there is an element in $B$. This element must be ${} \ge 0$ so that
$\idt{function{\dsh}norm}$ has the norm properties. Furthermore it
does not have to change the norm in all other cases, so it must be
$0$, as all other elements of $B$ are ${} \ge 0$.
Thus $B$ is defined as follows.
*};
constdefs
B :: "[ 'a set, 'a => real, 'a => real ] => real set"
"B V norm f ==
{z. z = 0r | (EX x:V. x ~= <0> & z = rabs (f x) * rinv (norm x))}";
text{* $\idt{function{\dsh}norm}$ is equal to the supremum of $B$,
if there exists a supremum. *};
constdefs
function_norm :: " ['a set, 'a => real, 'a => real] => real"
"function_norm V norm f == Sup UNIV (B V norm f)";
text{* $\idt{is{\dsh}function{\dsh}norm}$ also guarantees that there
is a funciton norm .*};
constdefs
is_function_norm ::
" ['a set, 'a => real, 'a => real] => real => bool"
"is_function_norm V norm f fn == is_Sup UNIV (B V norm f) fn";
lemma B_not_empty: "0r : B V norm f";
by (unfold B_def, force);
text {* The following lemma states every continous linearform on a
normed space $(V, \norm{\cdot})$ has a function norm. *};
lemma ex_fnorm [intro!!]:
"[| is_normed_vectorspace V norm; is_continous V norm f|]
==> is_function_norm V norm f (function_norm V norm f)";
proof (unfold function_norm_def is_function_norm_def
is_continous_def Sup_def, elim conjE, rule selectI2EX);
assume "is_normed_vectorspace V norm";
assume "is_linearform V f"
and e: "EX c. ALL x:V. rabs (f x) <= c * norm x";
txt {* The existence of the supremum is shown using the
completeness of the reals. Completeness means, that
for every non-empty and bounded set of reals there exists a
supremum. *};
show "EX a. is_Sup UNIV (B V norm f) a";
proof (unfold is_Sup_def, rule reals_complete);
txt {* First we have to show that $B$ is non-empty. *};
show "EX X. X : B V norm f";
proof (intro exI);
show "0r : (B V norm f)"; by (unfold B_def, force);
qed;
txt {* Then we have to show that $B$ is bounded. *};
from e; show "EX Y. isUb UNIV (B V norm f) Y";
proof;
txt {* We know that $f$ is bounded by some value $c$. *};
fix c; assume a: "ALL x:V. rabs (f x) <= c * norm x";
def b == "max c 0r";
show "?thesis";
proof (intro exI isUbI setleI ballI, unfold B_def,
elim CollectE disjE bexE conjE);
txt{* To proof the thesis, we have to show that there is
some constant b, which is greater than every $y$ in $B$.
Due to the definition of $B$ there are two cases for
$y\in B$. If $y = 0$ then $y$ is less than
$\idt{max}\ap c\ap 0$: *};
fix y; assume "y = 0r";
show "y <= b"; by (simp! add: le_max2);
txt{* The second case is
$y = \frac{|f\ap x|}{\norm x}$ for some
$x\in V$ with $x \neq \zero$. *};
next;
fix x y;
assume "x:V" "x ~= <0>"; (***
have ge: "0r <= rinv (norm x)";
by (rule real_less_imp_le, rule real_rinv_gt_zero,
rule normed_vs_norm_gt_zero); (***
proof (rule real_less_imp_le);
show "0r < rinv (norm x)";
proof (rule real_rinv_gt_zero);
show "0r < norm x"; ..;
qed;
qed; ***)
have nz: "norm x ~= 0r";
by (rule not_sym, rule lt_imp_not_eq,
rule normed_vs_norm_gt_zero); (***
proof (rule not_sym);
show "0r ~= norm x";
proof (rule lt_imp_not_eq);
show "0r < norm x"; ..;
qed;
qed; ***)***)
txt {* The thesis follows by a short calculation using the
fact that $f$ is bounded. *};
assume "y = rabs (f x) * rinv (norm x)";
also; have "... <= c * norm x * rinv (norm x)";
proof (rule real_mult_le_le_mono2);
show "0r <= rinv (norm x)";
by (rule real_less_imp_le, rule real_rinv_gt_zero,
rule normed_vs_norm_gt_zero);
from a; show "rabs (f x) <= c * norm x"; ..;
qed;
also; have "... = c * (norm x * rinv (norm x))";
by (rule real_mult_assoc);
also; have "(norm x * rinv (norm x)) = 1r";
proof (rule real_mult_inv_right);
show nz: "norm x ~= 0r";
by (rule not_sym, rule lt_imp_not_eq,
rule normed_vs_norm_gt_zero);
qed;
also; have "c * ... <= b "; by (simp! add: le_max1);
finally; show "y <= b"; .;
qed simp;
qed;
qed;
qed;
text {* The norm of a continous function is always $\geq 0$. *};
lemma fnorm_ge_zero [intro!!]:
"[| is_continous V norm f; is_normed_vectorspace V norm|]
==> 0r <= function_norm V norm f";
proof -;
assume c: "is_continous V norm f"
and n: "is_normed_vectorspace V norm";
txt {* The function norm is defined as the supremum of $B$.
So it is $\geq 0$ if all elements in $B$ are $\geq 0$, provided
the supremum exists and $B$ is not empty. *};
show ?thesis;
proof (unfold function_norm_def, rule sup_ub1);
show "ALL x:(B V norm f). 0r <= x";
proof (intro ballI, unfold B_def,
elim CollectE bexE conjE disjE);
fix x r;
assume "x : V" "x ~= <0>"
and r: "r = rabs (f x) * rinv (norm x)";
have ge: "0r <= rabs (f x)"; by (simp! only: rabs_ge_zero);
have "0r <= rinv (norm x)";
by (rule real_less_imp_le, rule real_rinv_gt_zero, rule);(***
proof (rule real_less_imp_le);
show "0r < rinv (norm x)";
proof (rule real_rinv_gt_zero);
show "0r < norm x"; ..;
qed;
qed; ***)
with ge; show "0r <= r";
by (simp only: r,rule real_le_mult_order);
qed (simp!);
txt {* Since $f$ is continous the function norm exists. *};
have "is_function_norm V norm f (function_norm V norm f)"; ..;
thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))";
by (unfold is_function_norm_def, unfold function_norm_def);
txt {* $B$ is non-empty by construction. *};
show "0r : B V norm f"; by (rule B_not_empty);
qed;
qed;
text{* The basic property of function norms is:
\begin{matharray}{l}
| f\ap x | \leq {\fnorm {f}} \cdot {\norm x}
\end{matharray}
*};
lemma norm_fx_le_norm_f_norm_x:
"[| is_normed_vectorspace V norm; x:V; is_continous V norm f |]
==> rabs (f x) <= (function_norm V norm f) * norm x";
proof -;
assume "is_normed_vectorspace V norm" "x:V"
and c: "is_continous V norm f";
have v: "is_vectorspace V"; ..;
assume "x:V";
txt{* The proof is by case analysis on $x$. *};
show ?thesis;
proof (rule case_split);
txt {* For the case $x = \zero$ the thesis follows
from the linearity of $f$: for every linear function
holds $f\ap \zero = 0$. *};
assume "x = <0>";
have "rabs (f x) = rabs (f <0>)"; by (simp!);
also; from v continous_linearform; have "f <0> = 0r"; ..;
also; note rabs_zero;
also; have "0r <= function_norm V norm f * norm x";
proof (rule real_le_mult_order);
show "0r <= function_norm V norm f"; ..;
show "0r <= norm x"; ..;
qed;
finally;
show "rabs (f x) <= function_norm V norm f * norm x"; .;
next;
assume "x ~= <0>";
have n: "0r <= norm x"; ..;
have nz: "norm x ~= 0r";
proof (rule lt_imp_not_eq [RS not_sym]);
show "0r < norm x"; ..;
qed;
txt {* For the case $x\neq \zero$ we derive the following
fact from the definition of the function norm:*};
have l: "rabs (f x) * rinv (norm x) <= function_norm V norm f";
proof (unfold function_norm_def, rule sup_ub);
from ex_fnorm [OF _ c];
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))";
by (simp! add: is_function_norm_def function_norm_def);
show "rabs (f x) * rinv (norm x) : B V norm f";
by (unfold B_def, intro CollectI disjI2 bexI [of _ x]
conjI, simp);
qed;
txt {* The thesis follows by a short calculation: *};
have "rabs (f x) = rabs (f x) * 1r"; by (simp!);
also; from nz; have "1r = rinv (norm x) * norm x";
by (rule real_mult_inv_left [RS sym]);
also;
have "rabs (f x) * ... = rabs (f x) * rinv (norm x) * norm x";
by (simp! add: real_mult_assoc [of "rabs (f x)"]);
also; have "... <= function_norm V norm f * norm x";
by (rule real_mult_le_le_mono2 [OF n l]);
finally;
show "rabs (f x) <= function_norm V norm f * norm x"; .;
qed;
qed;
text{* The function norm is the least positive real number for
which the inequation
\begin{matharray}{l}
| f\ap x | \leq c \cdot {\norm x}
\end{matharray}
holds.
*};
lemma fnorm_le_ub:
"[| is_normed_vectorspace V norm; is_continous V norm f;
ALL x:V. rabs (f x) <= c * norm x; 0r <= c |]
==> function_norm V norm f <= c";
proof (unfold function_norm_def);
assume "is_normed_vectorspace V norm";
assume c: "is_continous V norm f";
assume fb: "ALL x:V. rabs (f x) <= c * norm x"
and "0r <= c";
txt {* Suppose the inequation holds for some $c\geq 0$.
If $c$ is an upper bound of $B$, then $c$ is greater
than the function norm since the function norm is the
least upper bound.
*};
show "Sup UNIV (B V norm f) <= c";
proof (rule sup_le_ub);
from ex_fnorm [OF _ c];
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))";
by (simp! add: is_function_norm_def function_norm_def);
txt {* $c$ is an upper bound of $B$, i.~e.~every
$y\in B$ is less than $c$. *};
show "isUb UNIV (B V norm f) c";
proof (intro isUbI setleI ballI);
fix y; assume "y: B V norm f";
thus le: "y <= c";
proof (unfold B_def, elim CollectE disjE bexE conjE);
txt {* The first case for $y\in B$ is $y=0$. *};
assume "y = 0r";
show "y <= c"; by (force!);
txt{* The second case is
$y = \frac{|f\ap x|}{\norm x}$ for some
$x\in V$ with $x \neq \zero$. *};
next;
fix x;
assume "x : V" "x ~= <0>";
have lz: "0r < norm x";
by (simp! add: normed_vs_norm_gt_zero);
have nz: "norm x ~= 0r";
proof (rule not_sym);
from lz; show "0r ~= norm x";
by (simp! add: order_less_imp_not_eq);
qed;
from lz; have "0r < rinv (norm x)";
by (simp! add: real_rinv_gt_zero);
hence rinv_gez: "0r <= rinv (norm x)";
by (rule real_less_imp_le);
assume "y = rabs (f x) * rinv (norm x)";
also; from rinv_gez; have "... <= c * norm x * rinv (norm x)";
proof (rule real_mult_le_le_mono2);
show "rabs (f x) <= c * norm x"; by (rule bspec);
qed;
also; have "... <= c"; by (simp add: nz real_mult_assoc);
finally; show ?thesis; .;
qed;
qed force;
qed;
qed;
end;