(* Title: HOL/Real/HahnBanach/FunctionNorm.thy
ID: $Id$
Author: Gertrud Bauer, TU Munich
*)
header {* The norm of a function *}
theory FunctionNorm = NormedSpace + FunctionOrder:
subsection {* Continuous linear forms*}
text{* A linear form $f$ on a normed vector space $(V, \norm{\cdot})$
is \emph{continuous}, iff it is bounded, i.~e.
\[\Ex {c\in R}{\All {x\in V} {|f\ap x| \leq c \cdot \norm x}}\]
In our application no other functions than linear forms are considered,
so we can define continuous linear forms as bounded linear forms:
*}
constdefs
is_continuous ::
"['a::{plus, minus, zero} set, 'a => real, 'a => real] => bool"
"is_continuous V norm f ==
is_linearform V f \\<and> (\\<exists>c. \\<forall>x \\<in> V. |f x| <= c * norm x)"
lemma continuousI [intro]:
"[| is_linearform V f; !! x. x \\<in> V ==> |f x| <= c * norm x |]
==> is_continuous V norm f"
proof (unfold is_continuous_def, intro exI conjI ballI)
assume r: "!! x. x \\<in> V ==> |f x| <= c * norm x"
fix x assume "x \\<in> V" show "|f x| <= c * norm x" by (rule r)
qed
lemma continuous_linearform [intro?]:
"is_continuous V norm f ==> is_linearform V f"
by (unfold is_continuous_def) force
lemma continuous_bounded [intro?]:
"is_continuous V norm f
==> \\<exists>c. \\<forall>x \\<in> V. |f x| <= c * norm x"
by (unfold is_continuous_def) force
subsection{* The norm of a linear form *}
text{* The least real number $c$ for which holds
\[\All {x\in V}{|f\ap x| \leq c \cdot \norm x}\]
is called the \emph{norm} of $f$.
For non-trivial vector spaces $V \neq \{\zero\}$ the norm can be defined as
\[\fnorm {f} =\sup_{x\neq\zero}\frac{|f\ap x|}{\norm x} \]
For the case $V = \{\zero\}$ the supremum would be taken from an
empty set. Since $\bbbR$ 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 ${} \ge 0$ 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 are ${} \ge 0$.
Thus we define the set $B$ the supremum is taken from as
\[
\{ 0 \} \Union \left\{ \frac{|f\ap x|}{\norm x} \dt x\neq \zero \And x\in F\right\}
\]
*}
constdefs
B :: "[ 'a set, 'a => real, 'a::{plus, minus, zero} => real ] => real set"
"B V norm f ==
{#0} \\<union> {|f x| * rinv (norm x) | x. x \\<noteq> 0 \\<and> x \\<in> V}"
text{* $n$ is the function norm of $f$, iff
$n$ is the supremum of $B$.
*}
constdefs
is_function_norm ::
" ['a::{minus,plus,zero} => real, 'a set, 'a => real] => real => bool"
"is_function_norm f V norm fn == is_Sup UNIV (B V norm f) fn"
text{* $\idt{function{\dsh}norm}$ is equal to the supremum of $B$,
if the supremum exists. Otherwise it is undefined. *}
constdefs
function_norm :: " ['a::{minus,plus,zero} => real, 'a set, 'a => real] => real"
"function_norm f V norm == Sup UNIV (B V norm f)"
syntax
function_norm :: " ['a => real, 'a set, 'a => real] => real" ("\\<parallel>_\\<parallel>_,_")
lemma B_not_empty: "#0 \\<in> B V norm f"
by (unfold B_def, force)
text {* The following lemma states that every continuous linear form
on a normed space $(V, \norm{\cdot})$ has a function norm. *}
lemma ex_fnorm [intro?]:
"[| is_normed_vectorspace V norm; is_continuous V norm f|]
==> is_function_norm f V norm \\<parallel>f\\<parallel>V,norm"
proof (unfold function_norm_def is_function_norm_def
is_continuous_def Sup_def, elim conjE, rule selectI2EX)
assume "is_normed_vectorspace V norm"
assume "is_linearform V f"
and e: "\\<exists>c. \\<forall>x \\<in> V. |f x| <= c * norm x"
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. *}
show "\\<exists>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 "\\<exists>X. X \\<in> B V norm f"
proof (intro exI)
show "#0 \\<in> (B V norm f)" by (unfold B_def, force)
qed
txt {* Then we have to show that $B$ is bounded: *}
from e show "\\<exists>Y. isUb UNIV (B V norm f) Y"
proof
txt {* We know that $f$ is bounded by some value $c$. *}
fix c assume a: "\\<forall>x \\<in> V. |f x| <= c * norm x"
def b == "max c #0"
show "?thesis"
proof (intro exI isUbI setleI ballI, unfold B_def,
elim UnE CollectE exE conjE singletonE)
txt{* To proof the thesis, we have to show that there is
some constant $b$, such that $y \leq b$ for all $y\in B$.
Due to the definition of $B$ there are two cases for
$y\in B$. If $y = 0$ then $y \leq \idt{max}\ap c\ap 0$: *}
fix y assume "y = (#0::real)"
show "y <= b" by (simp! add: le_maxI2)
txt{* The second case is
$y = {|f\ap x|}/{\norm x}$ for some
$x\in V$ with $x \neq \zero$. *}
next
fix x y
assume "x \\<in> V" "x \\<noteq> 0" (***
have ge: "#0 <= 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 "#0 < rinv (norm x)";
proof (rule real_rinv_gt_zero);
show "#0 < norm x"; ..;
qed;
qed; *** )
have nz: "norm x \\<noteq> #0"
by (rule not_sym, rule lt_imp_not_eq,
rule normed_vs_norm_gt_zero) (***
proof (rule not_sym);
show "#0 \\<noteq> norm x";
proof (rule lt_imp_not_eq);
show "#0 < norm x"; ..;
qed;
qed; ***)***)
txt {* The thesis follows by a short calculation using the
fact that $f$ is bounded. *}
assume "y = |f x| * rinv (norm x)"
also have "... <= c * norm x * rinv (norm x)"
proof (rule real_mult_le_le_mono2)
show "#0 <= rinv (norm x)"
by (rule real_less_imp_le, rule real_rinv_gt_zero1,
rule normed_vs_norm_gt_zero)
from a show "|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)) = (#1::real)"
proof (rule real_mult_inv_right1)
show nz: "norm x \\<noteq> #0"
by (rule not_sym, rule lt_imp_not_eq,
rule normed_vs_norm_gt_zero)
qed
also have "c * ... <= b " by (simp! add: le_maxI1)
finally show "y <= b" .
qed simp
qed
qed
qed
text {* The norm of a continuous function is always $\geq 0$. *}
lemma fnorm_ge_zero [intro?]:
"[| is_continuous V norm f; is_normed_vectorspace V norm |]
==> #0 <= \\<parallel>f\\<parallel>V,norm"
proof -
assume c: "is_continuous 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 "\\<forall>x \\<in> (B V norm f). #0 <= x"
proof (intro ballI, unfold B_def,
elim UnE singletonE CollectE exE conjE)
fix x r
assume "x \\<in> V" "x \\<noteq> 0"
and r: "r = |f x| * rinv (norm x)"
have ge: "#0 <= |f x|" by (simp! only: abs_ge_zero)
have "#0 <= rinv (norm x)"
by (rule real_less_imp_le, rule real_rinv_gt_zero1, rule)(***
proof (rule real_less_imp_le);
show "#0 < rinv (norm x)";
proof (rule real_rinv_gt_zero);
show "#0 < norm x"; ..;
qed;
qed; ***)
with ge show "#0 <= r"
by (simp only: r, rule real_le_mult_order1a)
qed (simp!)
txt {* Since $f$ is continuous the function norm exists: *}
have "is_function_norm f V norm \\<parallel>f\\<parallel>V,norm" ..
thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"
by (unfold is_function_norm_def function_norm_def)
txt {* $B$ is non-empty by construction: *}
show "#0 \\<in> B V norm f" by (rule B_not_empty)
qed
qed
text{* \medskip The fundamental 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_continuous V norm f; is_normed_vectorspace V norm; x \\<in> V |]
==> |f x| <= \\<parallel>f\\<parallel>V,norm * norm x"
proof -
assume "is_normed_vectorspace V norm" "x \\<in> V"
and c: "is_continuous V norm f"
have v: "is_vectorspace V" ..
txt{* The proof is by case analysis on $x$. *}
show ?thesis
proof cases
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 "|f x| = |f 0|" by (simp!)
also from v continuous_linearform have "f 0 = #0" ..
also note abs_zero
also have "#0 <= \\<parallel>f\\<parallel>V,norm * norm x"
proof (rule real_le_mult_order1a)
show "#0 <= \\<parallel>f\\<parallel>V,norm" ..
show "#0 <= norm x" ..
qed
finally
show "|f x| <= \\<parallel>f\\<parallel>V,norm * norm x" .
next
assume "x \\<noteq> 0"
have n: "#0 < norm x" ..
hence nz: "norm x \\<noteq> #0"
by (simp only: lt_imp_not_eq)
txt {* For the case $x\neq \zero$ we derive the following
fact from the definition of the function norm:*}
have l: "|f x| * rinv (norm x) <= \\<parallel>f\\<parallel>V,norm"
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 "|f x| * rinv (norm x) \\<in> B V norm f"
by (unfold B_def, intro UnI2 CollectI exI [of _ x]
conjI, simp)
qed
txt {* The thesis now follows by a short calculation: *}
have "|f x| = |f x| * #1" by (simp!)
also from nz have "#1 = rinv (norm x) * norm x"
by (simp add: real_mult_inv_left1)
also have "|f x| * ... = |f x| * rinv (norm x) * norm x"
by (simp! add: real_mult_assoc)
also from n l have "... <= \\<parallel>f\\<parallel>V,norm * norm x"
by (simp add: real_mult_le_le_mono2)
finally show "|f x| <= \\<parallel>f\\<parallel>V,norm * norm x" .
qed
qed
text{* \medskip The function norm is the least positive real number for
which the following inequation holds:
\begin{matharray}{l}
| f\ap x | \leq c \cdot {\norm x}
\end{matharray}
*}
lemma fnorm_le_ub:
"[| is_continuous V norm f; is_normed_vectorspace V norm;
\\<forall>x \\<in> V. |f x| <= c * norm x; #0 <= c |]
==> \\<parallel>f\\<parallel>V,norm <= c"
proof (unfold function_norm_def)
assume "is_normed_vectorspace V norm"
assume c: "is_continuous V norm f"
assume fb: "\\<forall>x \\<in> V. |f x| <= c * norm x"
and "#0 <= 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 \\<in> B V norm f"
thus le: "y <= c"
proof (unfold B_def, elim UnE CollectE exE conjE singletonE)
txt {* The first case for $y\in B$ is $y=0$. *}
assume "y = #0"
show "y <= c" by (force!)
txt{* The second case is
$y = {|f\ap x|}/{\norm x}$ for some
$x\in V$ with $x \neq \zero$. *}
next
fix x
assume "x \\<in> V" "x \\<noteq> 0"
have lz: "#0 < norm x"
by (simp! add: normed_vs_norm_gt_zero)
have nz: "norm x \\<noteq> #0"
proof (rule not_sym)
from lz show "#0 \\<noteq> norm x"
by (simp! add: order_less_imp_not_eq)
qed
from lz have "#0 < rinv (norm x)"
by (simp! add: real_rinv_gt_zero1)
hence rinv_gez: "#0 <= rinv (norm x)"
by (rule real_less_imp_le)
assume "y = |f x| * rinv (norm x)"
also from rinv_gez have "... <= c * norm x * rinv (norm x)"
proof (rule real_mult_le_le_mono2)
show "|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