(* 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 @{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:
*}
constdefs
is_continuous ::
"'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
"is_continuous V norm f \<equiv>
is_linearform V f \<and> (\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x)"
lemma continuousI [intro]:
"is_linearform V f \<Longrightarrow> (\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * norm x)
\<Longrightarrow> is_continuous V norm f"
by (unfold is_continuous_def) blast
lemma continuous_linearform [intro?]:
"is_continuous V norm f \<Longrightarrow> is_linearform V f"
by (unfold is_continuous_def) blast
lemma continuous_bounded [intro?]:
"is_continuous V norm f
\<Longrightarrow> \<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
by (unfold is_continuous_def) blast
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 function_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 of are @{text "{} \<ge> 0"}.
Thus we define the set @{text B} the supremum is taken from as
\begin{center}
@{text "{0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel>. x \<noteq> 0 \<and> x \<in> F}"}
\end{center}
*}
constdefs
B :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a::{plus, minus, zero} \<Rightarrow> real) \<Rightarrow> real set"
"B V norm f \<equiv>
{0} \<union> {\<bar>f x\<bar> * inverse (norm x) | x. x \<noteq> 0 \<and> x \<in> V}"
text {*
@{text n} is the function norm of @{text f}, iff @{text n} is the
supremum of @{text B}.
*}
constdefs
is_function_norm ::
"('a::{minus,plus,zero} \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool"
"is_function_norm f V norm fn \<equiv> is_Sup UNIV (B V norm f) fn"
text {*
@{text function_norm} is equal to the supremum of @{text B}, if the
supremum exists. Otherwise it is undefined.
*}
constdefs
function_norm :: "('a::{minus,plus,zero} \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real"
"function_norm f V norm \<equiv> Sup UNIV (B V norm f)"
syntax
function_norm :: "('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("\<parallel>_\<parallel>_,_")
lemma B_not_empty: "0 \<in> B V norm f"
by (unfold B_def) blast
text {*
The following lemma states that every continuous linear form on a
normed space @{text "(V, \<parallel>\<cdot>\<parallel>)"} has a function norm.
*}
lemma ex_fnorm [intro?]:
"is_normed_vectorspace V norm \<Longrightarrow> is_continuous V norm f
\<Longrightarrow> 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 someI2_ex)
assume "is_normed_vectorspace V norm"
assume "is_linearform V f"
and e: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> 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 @{text B} is non-empty: *}
show "\<exists>X. X \<in> B V norm f"
proof
show "0 \<in> (B V norm f)" by (unfold B_def) blast
qed
txt {* Then we have to show that @{text B} is bounded: *}
from e show "\<exists>Y. isUb UNIV (B V norm f) Y"
proof
txt {* We know that @{text f} is bounded by some value @{text c}. *}
fix c assume a: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
def b \<equiv> "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 @{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 for @{text "y \<in> B"}. If @{text "y = 0"} then
@{text "y \<le> max c 0"}: *}
fix y assume "y = (0::real)"
show "y \<le> b" by (simp! add: le_maxI2)
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"}. *}
next
fix x y
assume "x \<in> V" "x \<noteq> 0"
txt {* The thesis follows by a short calculation using the
fact that @{text f} is bounded. *}
assume "y = \<bar>f x\<bar> * inverse (norm x)"
also have "... \<le> c * norm x * inverse (norm x)"
proof (rule real_mult_le_le_mono2)
show "0 \<le> inverse (norm x)"
by (rule order_less_imp_le, rule real_inverse_gt_0,
rule normed_vs_norm_gt_zero)
from a show "\<bar>f x\<bar> \<le> c * norm x" ..
qed
also have "... = c * (norm x * inverse (norm x))"
by (rule real_mult_assoc)
also have "(norm x * inverse (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 * ... \<le> b " by (simp! add: le_maxI1)
finally show "y \<le> b" .
qed simp
qed
qed
qed
text {* The norm of a continuous function is always @{text "\<ge> 0"}. *}
lemma fnorm_ge_zero [intro?]:
"is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm
\<Longrightarrow> 0 \<le> \<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 @{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. *}
show ?thesis
proof (unfold function_norm_def, rule sup_ub1)
show "\<forall>x \<in> (B V norm f). 0 \<le> 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 = \<bar>f x\<bar> * inverse (norm x)"
have ge: "0 \<le> \<bar>f x\<bar>" by (simp! only: abs_ge_zero)
have "0 \<le> inverse (norm x)"
by (rule order_less_imp_le, rule real_inverse_gt_0, rule)(***
proof (rule order_less_imp_le);
show "0 < inverse (norm x)";
proof (rule real_inverse_gt_zero);
show "0 < norm x"; ..;
qed;
qed; ***)
with ge show "0 \<le> r"
by (simp only: r, rule real_le_mult_order1a)
qed (simp!)
txt {* Since @{text 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 {* @{text 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{center}
@{text "\<bar>f x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
\end{center}
*}
lemma norm_fx_le_norm_f_norm_x:
"is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm \<Longrightarrow> x \<in> V
\<Longrightarrow> \<bar>f x\<bar> \<le> \<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 @{text x}. *}
show ?thesis
proof cases
txt {* For the case @{text "x = 0"} the thesis follows from the
linearity of @{text f}: for every linear function holds
@{text "f 0 = 0"}. *}
assume "x = 0"
have "\<bar>f x\<bar> = \<bar>f 0\<bar>" by (simp!)
also from v continuous_linearform have "f 0 = 0" ..
also note abs_zero
also have "0 \<le> \<parallel>f\<parallel>V,norm * norm x"
proof (rule real_le_mult_order1a)
show "0 \<le> \<parallel>f\<parallel>V,norm" ..
show "0 \<le> norm x" ..
qed
finally
show "\<bar>f x\<bar> \<le> \<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 @{text "x \<noteq> 0"} we derive the following fact
from the definition of the function norm:*}
have l: "\<bar>f x\<bar> * inverse (norm x) \<le> \<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 "\<bar>f x\<bar> * inverse (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 "\<bar>f x\<bar> = \<bar>f x\<bar> * 1" by (simp!)
also from nz have "1 = inverse (norm x) * norm x"
by (simp add: real_mult_inv_left1)
also have "\<bar>f x\<bar> * ... = \<bar>f x\<bar> * inverse (norm x) * norm x"
by (simp! add: real_mult_assoc)
also from n l have "... \<le> \<parallel>f\<parallel>V,norm * norm x"
by (simp add: real_mult_le_le_mono2)
finally show "\<bar>f x\<bar> \<le> \<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{center}
@{text "\<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
*}
lemma fnorm_le_ub:
"is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm \<Longrightarrow>
\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x \<Longrightarrow> 0 \<le> c
\<Longrightarrow> \<parallel>f\<parallel>V,norm \<le> 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. \<bar>f x\<bar> \<le> c * norm x"
and "0 \<le> c"
txt {* Suppose the inequation holds for some @{text "c \<ge> 0"}. If
@{text c} is an upper bound of @{text B}, then @{text c} is greater
than the function norm since the function norm is the least upper
bound. *}
show "Sup UNIV (B V norm f) \<le> 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 {* @{text c} is an upper bound of @{text B}, i.e. every
@{text "y \<in> B"} is less than @{text 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 \<le> c"
proof (unfold B_def, elim UnE CollectE exE conjE singletonE)
txt {* The first case for @{text "y \<in> B"} is @{text "y = 0"}. *}
assume "y = 0"
show "y \<le> c" by (blast!)
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"}. *}
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 < inverse (norm x)"
by (simp! add: real_inverse_gt_0)
hence inverse_gez: "0 \<le> inverse (norm x)"
by (rule order_less_imp_le)
assume "y = \<bar>f x\<bar> * inverse (norm x)"
also from inverse_gez have "... \<le> c * norm x * inverse (norm x)"
proof (rule real_mult_le_le_mono2)
show "\<bar>f x\<bar> \<le> c * norm x" by (rule bspec)
qed
also have "... \<le> c" by (simp add: nz real_mult_assoc)
finally show ?thesis .
qed
qed blast
qed
qed
end