author | wenzelm |
Fri, 22 Oct 1999 20:14:31 +0200 | |
changeset 7917 | 5e5b9813cce7 |
parent 7808 | fd019ac3485f |
child 7927 | b50446a33c16 |
permissions | -rw-r--r-- |
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(* Title: HOL/Real/HahnBanach/FunctionNorm.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, TU Munich |
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*) |
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header {* The norm of a function *}; |
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theory FunctionNorm = NormedSpace + FunctionOrder:; |
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subsection {* Continous linearforms*}; |
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text{* A linearform $f$ on a normed vector space $(V, \norm{\cdot})$ |
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is \emph{continous}, iff it is bounded, i.~e. |
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\[\exists\ap c\in R.\ap \forall\ap x\in V.\ap |
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|f\ap x| \leq c \cdot \norm x.\] |
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In our application no other functions than linearforms are considered, |
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so we can define continous linearforms as follows: |
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*}; |
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constdefs |
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is_continous :: |
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"['a::{minus, plus} set, 'a => real, 'a => real] => bool" |
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"is_continous V norm f == |
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is_linearform V f & (EX c. ALL x:V. rabs (f x) <= c * norm x)"; |
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lemma continousI [intro]: |
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"[| is_linearform V f; !! x. x:V ==> rabs (f x) <= c * norm x |] |
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==> is_continous V norm f"; |
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proof (unfold is_continous_def, intro exI conjI ballI); |
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assume r: "!! x. x:V ==> rabs (f x) <= c * norm x"; |
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fix x; assume "x:V"; show "rabs (f x) <= c * norm x"; by (rule r); |
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qed; |
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lemma continous_linearform [intro!!]: |
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"is_continous V norm f ==> is_linearform V f"; |
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by (unfold is_continous_def) force; |
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lemma continous_bounded [intro!!]: |
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"is_continous V norm f |
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==> EX c. ALL x:V. rabs (f x) <= c * norm x"; |
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by (unfold is_continous_def) force; |
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subsection{* The norm of a linearform *}; |
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text{* The least real number $c$ for which holds |
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\[\forall\ap x\in V.\ap |f\ap x| \leq c \cdot \norm x\] |
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is called the \emph{norm} of $f$. |
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For the non-trivial vector space $V$ the norm can be defined as |
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\[\fnorm {f} =\sup_{x\neq\zero}\frac{|f\ap x|}{\norm x}. \] |
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For the case that the vector space $V$ contains only the zero vector |
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set, the set $B$ this supremum is taken from would be empty, and any |
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real number is a supremum of $B$. So it must be guarateed that there |
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is an element in $B$. This element must be greater or equal $0$ so |
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that $\idt{function{\dsh}norm}$ has the norm properties. Furthermore |
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it does not have to change the norm in all other cases, so it must be |
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$0$, as all other elements of $B$ are greater or equal $0$. |
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Thus $B$ is defined as follows. |
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*}; |
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constdefs |
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B :: "[ 'a set, 'a => real, 'a => real ] => real set" |
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"B V norm f == |
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{z. z = 0r | (EX x:V. x ~= <0> & z = rabs (f x) * rinv (norm x))}"; |
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text{* $\idt{function{\dsh}norm}$ is equal to the supremum of $B$, |
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if there exists a supremum. *}; |
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constdefs |
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function_norm :: " ['a set, 'a => real, 'a => real] => real" |
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"function_norm V norm f == Sup UNIV (B V norm f)"; |
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text{* $\idt{is{\dsh}function{\dsh}norm}$ also guarantees that there |
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is a funciton norm .*}; |
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constdefs |
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is_function_norm :: |
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" ['a set, 'a => real, 'a => real] => real => bool" |
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"is_function_norm V norm f fn == is_Sup UNIV (B V norm f) fn"; |
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lemma B_not_empty: "0r : B V norm f"; |
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by (unfold B_def, force); |
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text {* The following lemma states every continous linearform on a |
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normed space $(V, \norm{\cdot})$ has a function norm. *}; |
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lemma ex_fnorm [intro!!]: |
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"[| is_normed_vectorspace V norm; is_continous V norm f|] |
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==> is_function_norm V norm f (function_norm V norm f)"; |
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proof (unfold function_norm_def is_function_norm_def |
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is_continous_def Sup_def, elim conjE, rule selectI2EX); |
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assume "is_normed_vectorspace V norm"; |
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assume "is_linearform V f" |
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and e: "EX c. ALL x:V. rabs (f x) <= c * norm x"; |
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txt {* The existence of the supremum is shown using the |
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completeness of the reals. Completeness means, that |
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for every non-empty and bounded set of reals there exists a |
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supremum. *}; |
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show "EX a. is_Sup UNIV (B V norm f) a"; |
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proof (unfold is_Sup_def, rule reals_complete); |
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txt {* First we have to show that $B$ is non-empty. *}; |
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show "EX X. X : B V norm f"; |
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proof (intro exI); |
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show "0r : (B V norm f)"; by (unfold B_def, force); |
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qed; |
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txt {* Then we have to show that $B$ is bounded. *}; |
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from e; show "EX Y. isUb UNIV (B V norm f) Y"; |
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proof; |
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txt {* We know that $f$ is bounded by some value $c$. *}; |
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fix c; assume a: "ALL x:V. rabs (f x) <= c * norm x"; |
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def b == "max c 0r"; |
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show "?thesis"; |
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proof (intro exI isUbI setleI ballI, unfold B_def, |
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elim CollectE disjE bexE conjE); |
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txt{* To proof the thesis, we have to show that there is |
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some constant b, which is greater than every $y$ in $B$. |
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Due to the definition of $B$ there are two cases for |
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$y\in B$. If $y = 0$ then $y$ is less than |
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$\idt{max}\ap c\ap 0$: *}; |
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fix y; assume "y = 0r"; |
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show "y <= b"; by (simp! add: le_max2); |
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txt{* The second case is |
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$y = \frac{|f\ap x|}{\norm x}$ for some |
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$x\in V$ with $x \neq \zero$. *}; |
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next; |
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fix x y; |
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assume "x:V" "x ~= <0>"; (*** |
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have ge: "0r <= rinv (norm x)"; |
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by (rule real_less_imp_le, rule real_rinv_gt_zero, |
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rule normed_vs_norm_gt_zero); (*** |
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proof (rule real_less_imp_le); |
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show "0r < rinv (norm x)"; |
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proof (rule real_rinv_gt_zero); |
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show "0r < norm x"; ..; |
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qed; |
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qed; ***) |
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have nz: "norm x ~= 0r"; |
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by (rule not_sym, rule lt_imp_not_eq, |
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rule normed_vs_norm_gt_zero); (*** |
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proof (rule not_sym); |
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show "0r ~= norm x"; |
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proof (rule lt_imp_not_eq); |
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show "0r < norm x"; ..; |
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qed; |
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qed; ***)***) |
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txt {* The thesis follows by a short calculation using the |
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fact that $f$ is bounded. *}; |
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assume "y = rabs (f x) * rinv (norm x)"; |
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also; have "... <= c * norm x * rinv (norm x)"; |
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proof (rule real_mult_le_le_mono2); |
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show "0r <= rinv (norm x)"; |
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by (rule real_less_imp_le, rule real_rinv_gt_zero, |
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rule normed_vs_norm_gt_zero); |
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from a; show "rabs (f x) <= c * norm x"; ..; |
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qed; |
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also; have "... = c * (norm x * rinv (norm x))"; |
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by (rule real_mult_assoc); |
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also; have "(norm x * rinv (norm x)) = 1r"; |
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proof (rule real_mult_inv_right); |
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show nz: "norm x ~= 0r"; |
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by (rule not_sym, rule lt_imp_not_eq, |
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rule normed_vs_norm_gt_zero); |
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qed; |
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also; have "c * ... <= b "; by (simp! add: le_max1); |
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finally; show "y <= b"; .; |
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qed simp; |
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qed; |
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qed; |
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187 |
qed; |
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text {* The norm of a continous function is always $\geq 0$. *}; |
190 |
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7808 | 191 |
lemma fnorm_ge_zero [intro!!]: |
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"[| is_continous V norm f; is_normed_vectorspace V norm|] |
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==> 0r <= function_norm V norm f"; |
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194 |
proof -; |
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assume c: "is_continous V norm f" |
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and n: "is_normed_vectorspace V norm"; |
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197 |
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txt {* The function norm is defined as the supremum of $B$. |
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So it is $\geq 0$ if all elements in $B$ are $\geq 0$, provided |
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the supremum exists and $B$ is not empty. *}; |
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show ?thesis; |
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proof (unfold function_norm_def, rule sup_ub1); |
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204 |
show "ALL x:(B V norm f). 0r <= x"; |
7917 | 205 |
proof (intro ballI, unfold B_def, |
206 |
elim CollectE bexE conjE disjE); |
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fix x r; |
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assume "x : V" "x ~= <0>" |
|
209 |
and r: "r = rabs (f x) * rinv (norm x)"; |
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210 |
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211 |
have ge: "0r <= rabs (f x)"; by (simp! only: rabs_ge_zero); |
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have "0r <= rinv (norm x)"; |
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213 |
by (rule real_less_imp_le, rule real_rinv_gt_zero, rule);(*** |
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proof (rule real_less_imp_le); |
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show "0r < rinv (norm x)"; |
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proof (rule real_rinv_gt_zero); |
217 |
show "0r < norm x"; ..; |
|
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qed; |
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qed; ***) |
220 |
with ge; show "0r <= r"; |
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by (simp only: r,rule real_le_mult_order); |
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qed (simp!); |
7917 | 223 |
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txt {* Since $f$ is continous the function norm exists. *}; |
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225 |
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226 |
have "is_function_norm V norm f (function_norm V norm f)"; ..; |
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227 |
thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
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by (unfold is_function_norm_def, unfold function_norm_def); |
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txt {* $B$ is non-empty by construction. *}; |
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show "0r : B V norm f"; by (rule B_not_empty); |
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qed; |
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234 |
qed; |
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7917 | 236 |
text{* The basic property of function norms is: |
237 |
\begin{matharray}{l} |
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238 |
| f\ap x | \leq {\fnorm {f}} \cdot {\norm x} |
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239 |
\end{matharray} |
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*}; |
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241 |
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lemma norm_fx_le_norm_f_norm_x: |
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"[| is_normed_vectorspace V norm; x:V; is_continous V norm f |] |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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changeset
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244 |
==> rabs (f x) <= (function_norm V norm f) * norm x"; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
245 |
proof -; |
7917 | 246 |
assume "is_normed_vectorspace V norm" "x:V" |
247 |
and c: "is_continous V norm f"; |
|
7566 | 248 |
have v: "is_vectorspace V"; ..; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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249 |
assume "x:V"; |
7917 | 250 |
|
251 |
txt{* The proof is by case analysis on $x$. *}; |
|
252 |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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253 |
show "?thesis"; |
7917 | 254 |
proof (rule case_split); |
255 |
||
256 |
txt {* For the case $x = \zero$ the thesis follows |
|
257 |
from the linearity of $f$: for every linear function |
|
258 |
holds $f\ap \zero = 0$. *}; |
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259 |
||
260 |
assume "x = <0>"; |
|
261 |
have "rabs (f x) = rabs (f <0>)"; by (simp!); |
|
262 |
also; from v continous_linearform; have "f <0> = 0r"; ..; |
|
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also; note rabs_zero; |
|
264 |
also; have "0r <= function_norm V norm f * norm x"; |
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265 |
proof (rule real_le_mult_order); |
|
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show "0r <= function_norm V norm f"; ..; |
|
267 |
show "0r <= norm x"; ..; |
|
268 |
qed; |
|
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finally; |
|
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show "rabs (f x) <= function_norm V norm f * norm x"; .; |
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271 |
||
272 |
next; |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
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diff
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273 |
assume "x ~= <0>"; |
7917 | 274 |
have n: "0r <= norm x"; ..; |
275 |
have nz: "norm x ~= 0r"; |
|
276 |
proof (rule lt_imp_not_eq [RS not_sym]); |
|
277 |
show "0r < norm x"; ..; |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
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278 |
qed; |
7917 | 279 |
|
280 |
txt {* For the case $x\neq \zero$ we derive the following |
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fact from the definition of the function norm:*}; |
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282 |
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283 |
have l: "rabs (f x) * rinv (norm x) <= function_norm V norm f"; |
|
284 |
proof (unfold function_norm_def, rule sup_ub); |
|
285 |
from ex_fnorm [OF _ c]; |
|
286 |
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
|
287 |
by (simp! add: is_function_norm_def function_norm_def); |
|
288 |
show "rabs (f x) * rinv (norm x) : B V norm f"; |
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by (unfold B_def, intro CollectI disjI2 bexI [of _ x] |
|
290 |
conjI, simp); |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
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291 |
qed; |
7917 | 292 |
|
293 |
txt {* The thesis follows by a short calculation: *}; |
|
294 |
||
295 |
have "rabs (f x) = rabs (f x) * 1r"; by (simp!); |
|
296 |
also; from nz; have "1r = rinv (norm x) * norm x"; |
|
297 |
by (rule real_mult_inv_left [RS sym]); |
|
298 |
also; |
|
299 |
have "rabs (f x) * ... = rabs (f x) * rinv (norm x) * norm x"; |
|
300 |
by (simp! add: real_mult_assoc [of "rabs (f x)"]); |
|
301 |
also; have "... <= function_norm V norm f * norm x"; |
|
302 |
by (rule real_mult_le_le_mono2 [OF n l]); |
|
303 |
finally; |
|
304 |
show "rabs (f x) <= function_norm V norm f * norm x"; .; |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
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|
305 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
306 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
307 |
|
7917 | 308 |
text{* The function norm is the least positive real number for |
309 |
which the inequation |
|
310 |
\begin{matharray}{l} |
|
311 |
| f\ap x | \leq c \cdot {\norm x} |
|
312 |
\end{matharray} |
|
313 |
holds. |
|
314 |
*}; |
|
315 |
||
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
316 |
lemma fnorm_le_ub: |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
317 |
"[| is_normed_vectorspace V norm; is_continous V norm f; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
318 |
ALL x:V. rabs (f x) <= c * norm x; 0r <= c |] |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
319 |
==> function_norm V norm f <= c"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
320 |
proof (unfold function_norm_def); |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
321 |
assume "is_normed_vectorspace V norm"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
322 |
assume c: "is_continous V norm f"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
323 |
assume fb: "ALL x:V. rabs (f x) <= c * norm x" |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
324 |
and "0r <= c"; |
7917 | 325 |
|
326 |
txt {* Suppose the inequation holds for some $c\geq 0$. |
|
327 |
If $c$ is an upper bound of $B$, then $c$ is greater |
|
328 |
than the function norm since the function norm is the |
|
329 |
least upper bound. |
|
330 |
*}; |
|
331 |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
332 |
show "Sup UNIV (B V norm f) <= c"; |
7656 | 333 |
proof (rule sup_le_ub); |
7808 | 334 |
from ex_fnorm [OF _ c]; |
335 |
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
|
7566 | 336 |
by (simp! add: is_function_norm_def function_norm_def); |
7917 | 337 |
|
338 |
txt {* $c$ is an upper bound of $B$, i.~e.~every |
|
339 |
$y\in B$ is less than $c$. *}; |
|
340 |
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599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
341 |
show "isUb UNIV (B V norm f) c"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
342 |
proof (intro isUbI setleI ballI); |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
343 |
fix y; assume "y: B V norm f"; |
7566 | 344 |
thus le: "y <= c"; |
7917 | 345 |
proof (unfold B_def, elim CollectE disjE bexE conjE); |
346 |
||
347 |
txt {* The first case for $y\in B$ is $y=0$. *}; |
|
348 |
||
349 |
assume "y = 0r"; |
|
350 |
show "y <= c"; by (force!); |
|
351 |
||
352 |
txt{* The second case is |
|
353 |
$y = \frac{|f\ap x|}{\norm x}$ for some |
|
354 |
$x\in V$ with $x \neq \zero$. *}; |
|
355 |
||
356 |
next; |
|
357 |
fix x; |
|
358 |
assume "x : V" "x ~= <0>"; |
|
359 |
||
360 |
have lz: "0r < norm x"; |
|
361 |
by (simp! add: normed_vs_norm_gt_zero); |
|
7566 | 362 |
|
7917 | 363 |
have nz: "norm x ~= 0r"; |
7566 | 364 |
proof (rule not_sym); |
7917 | 365 |
from lz; show "0r ~= norm x"; |
366 |
by (simp! add: order_less_imp_not_eq); |
|
7566 | 367 |
qed; |
368 |
||
7917 | 369 |
from lz; have "0r < rinv (norm x)"; |
7566 | 370 |
by (simp! add: real_rinv_gt_zero); |
7917 | 371 |
hence rinv_gez: "0r <= rinv (norm x)"; |
7808 | 372 |
by (rule real_less_imp_le); |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
373 |
|
7917 | 374 |
assume "y = rabs (f x) * rinv (norm x)"; |
375 |
also; from rinv_gez; have "... <= c * norm x * rinv (norm x)"; |
|
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
376 |
proof (rule real_mult_le_le_mono2); |
7917 | 377 |
show "rabs (f x) <= c * norm x"; by (rule bspec); |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
378 |
qed; |
7917 | 379 |
also; have "... <= c"; by (simp add: nz real_mult_assoc); |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
380 |
finally; show ?thesis; .; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
381 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
382 |
qed force; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
383 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
384 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
385 |
|
7808 | 386 |
end; |