author | fleuriot |
Thu, 01 Jun 2000 11:22:27 +0200 | |
changeset 9013 | 9dd0274f76af |
parent 8838 | 4eaa99f0d223 |
child 9035 | 371f023d3dbd |
permissions | -rw-r--r-- |
7566 | 1 |
(* 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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7978 | 10 |
subsection {* Continuous linear forms*}; |
7917 | 11 |
|
7978 | 12 |
text{* A linear form $f$ on a normed vector space $(V, \norm{\cdot})$ |
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is \emph{continuous}, iff it is bounded, i.~e. |
|
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\[\Ex {c\in R}{\All {x\in V} {|f\ap x| \leq c \cdot \norm x}}\] |
|
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In our application no other functions than linear forms are considered, |
|
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so we can define continuous linear forms as bounded linear forms: |
|
7917 | 17 |
*}; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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19 |
constdefs |
7978 | 20 |
is_continuous :: |
7917 | 21 |
"['a::{minus, plus} set, 'a => real, 'a => real] => bool" |
7978 | 22 |
"is_continuous V norm f == |
8838 | 23 |
is_linearform V f & (EX c. ALL x:V. abs (f x) <= c * norm x)"; |
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|
7978 | 25 |
lemma continuousI [intro]: |
8838 | 26 |
"[| is_linearform V f; !! x. x:V ==> abs (f x) <= c * norm x |] |
7978 | 27 |
==> is_continuous V norm f"; |
28 |
proof (unfold is_continuous_def, intro exI conjI ballI); |
|
8838 | 29 |
assume r: "!! x. x:V ==> abs (f x) <= c * norm x"; |
30 |
fix x; assume "x:V"; show "abs (f x) <= c * norm x"; by (rule r); |
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qed; |
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33 |
lemma continuous_linearform [intro??]: |
7978 | 34 |
"is_continuous V norm f ==> is_linearform V f"; |
35 |
by (unfold is_continuous_def) force; |
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lemma continuous_bounded [intro??]: |
7978 | 38 |
"is_continuous V norm f |
8838 | 39 |
==> EX c. ALL x:V. abs (f x) <= c * norm x"; |
7978 | 40 |
by (unfold is_continuous_def) force; |
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|
7978 | 42 |
subsection{* The norm of a linear form *}; |
7917 | 43 |
|
44 |
||
45 |
text{* The least real number $c$ for which holds |
|
7978 | 46 |
\[\All {x\in V}{|f\ap x| \leq c \cdot \norm x}\] |
7917 | 47 |
is called the \emph{norm} of $f$. |
48 |
||
7978 | 49 |
For non-trivial vector spaces $V \neq \{\zero\}$ the norm can be defined as |
7927 | 50 |
\[\fnorm {f} =\sup_{x\neq\zero}\frac{|f\ap x|}{\norm x} \] |
7917 | 51 |
|
7978 | 52 |
For the case $V = \{\zero\}$ the supremum would be taken from an |
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empty set. Since $\bbbR$ is unbounded, there would be no supremum. To |
|
54 |
avoid this situation it must be guaranteed that there is an element in |
|
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this set. This element must be ${} \ge 0$ so that |
|
7927 | 56 |
$\idt{function{\dsh}norm}$ has the norm properties. Furthermore it |
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does not have to change the norm in all other cases, so it must be |
|
7978 | 58 |
$0$, as all other elements of are ${} \ge 0$. |
7917 | 59 |
|
7978 | 60 |
Thus we define the set $B$ the supremum is taken from as |
61 |
\[ |
|
62 |
\{ 0 \} \Un \left\{ \frac{|f\ap x|}{\norm x} \dt x\neq \zero \And x\in F\right\} |
|
63 |
\] |
|
7917 | 64 |
*}; |
65 |
||
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constdefs |
7917 | 67 |
B :: "[ 'a set, 'a => real, 'a => real ] => real set" |
7808 | 68 |
"B V norm f == |
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{(#0::real)} \Un {abs (f x) * rinv (norm x) | x. x ~= 00 & x:V}"; |
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|
7978 | 71 |
text{* $n$ is the function norm of $f$, iff |
72 |
$n$ is the supremum of $B$. |
|
73 |
*}; |
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74 |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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75 |
constdefs |
7917 | 76 |
is_function_norm :: |
77 |
" ['a set, 'a => real, 'a => real] => real => bool" |
|
78 |
"is_function_norm V norm f fn == is_Sup UNIV (B V norm f) fn"; |
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79 |
|
7978 | 80 |
text{* $\idt{function{\dsh}norm}$ is equal to the supremum of $B$, |
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if the supremum exists. Otherwise it is undefined. *}; |
|
82 |
||
83 |
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)"; |
|
86 |
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lemma B_not_empty: "(#0::real) : B V norm f"; |
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by (unfold B_def, force); |
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89 |
|
7978 | 90 |
text {* The following lemma states that every continuous linear form |
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on a normed space $(V, \norm{\cdot})$ has a function norm. *}; |
|
7917 | 92 |
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lemma ex_fnorm [intro??]: |
7978 | 94 |
"[| is_normed_vectorspace V norm; is_continuous V norm f|] |
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95 |
==> is_function_norm V norm f (function_norm V norm f)"; |
7917 | 96 |
proof (unfold function_norm_def is_function_norm_def |
7978 | 97 |
is_continuous_def Sup_def, elim conjE, rule selectI2EX); |
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assume "is_normed_vectorspace V norm"; |
7808 | 99 |
assume "is_linearform V f" |
8838 | 100 |
and e: "EX c. ALL x:V. abs (f x) <= c * norm x"; |
7917 | 101 |
|
102 |
txt {* The existence of the supremum is shown using the |
|
103 |
completeness of the reals. Completeness means, that |
|
7978 | 104 |
every non-empty bounded set of reals has a |
7917 | 105 |
supremum. *}; |
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106 |
show "EX a. is_Sup UNIV (B V norm f) a"; |
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107 |
proof (unfold is_Sup_def, rule reals_complete); |
7917 | 108 |
|
7978 | 109 |
txt {* First we have to show that $B$ is non-empty: *}; |
7917 | 110 |
|
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show "EX X. X : B V norm f"; |
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proof (intro exI); |
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113 |
show "(#0::real) : (B V norm f)"; by (unfold B_def, force); |
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114 |
qed; |
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115 |
|
7978 | 116 |
txt {* Then we have to show that $B$ is bounded: *}; |
7917 | 117 |
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from e; show "EX Y. isUb UNIV (B V norm f) Y"; |
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119 |
proof; |
7917 | 120 |
|
121 |
txt {* We know that $f$ is bounded by some value $c$. *}; |
|
122 |
||
8838 | 123 |
fix c; assume a: "ALL x:V. abs (f x) <= c * norm x"; |
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def b == "max c (#0::real)"; |
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125 |
|
7917 | 126 |
show "?thesis"; |
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proof (intro exI isUbI setleI ballI, unfold B_def, |
7978 | 128 |
elim UnE CollectE exE conjE singletonE); |
7917 | 129 |
|
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txt{* To proof the thesis, we have to show that there is |
|
7978 | 131 |
some constant $b$, such that $y \leq b$ for all $y\in B$. |
7917 | 132 |
Due to the definition of $B$ there are two cases for |
7978 | 133 |
$y\in B$. If $y = 0$ then $y \leq idt{max}\ap c\ap 0$: *}; |
7917 | 134 |
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fix y; assume "y = (#0::real)"; |
7917 | 136 |
show "y <= b"; by (simp! add: le_max2); |
137 |
||
138 |
txt{* The second case is |
|
7978 | 139 |
$y = {|f\ap x|}/{\norm x}$ for some |
7917 | 140 |
$x\in V$ with $x \neq \zero$. *}; |
141 |
||
142 |
next; |
|
143 |
fix x y; |
|
8703 | 144 |
assume "x:V" "x ~= 00"; (*** |
7917 | 145 |
|
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146 |
have ge: "(#0::real) <= rinv (norm x)"; |
7917 | 147 |
by (rule real_less_imp_le, rule real_rinv_gt_zero, |
148 |
rule normed_vs_norm_gt_zero); (*** |
|
7656 | 149 |
proof (rule real_less_imp_le); |
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150 |
show "(#0::real) < rinv (norm x)"; |
7566 | 151 |
proof (rule real_rinv_gt_zero); |
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152 |
show "(#0::real) < norm x"; ..; |
7566 | 153 |
qed; |
7917 | 154 |
qed; ***) |
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155 |
have nz: "norm x ~= (#0::real)"; |
7917 | 156 |
by (rule not_sym, rule lt_imp_not_eq, |
157 |
rule normed_vs_norm_gt_zero); (*** |
|
158 |
proof (rule not_sym); |
|
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159 |
show "(#0::real) ~= norm x"; |
7917 | 160 |
proof (rule lt_imp_not_eq); |
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161 |
show "(#0::real) < norm x"; ..; |
7917 | 162 |
qed; |
163 |
qed; ***)***) |
|
164 |
||
165 |
txt {* The thesis follows by a short calculation using the |
|
166 |
fact that $f$ is bounded. *}; |
|
167 |
||
8838 | 168 |
assume "y = abs (f x) * rinv (norm x)"; |
7917 | 169 |
also; have "... <= c * norm x * rinv (norm x)"; |
170 |
proof (rule real_mult_le_le_mono2); |
|
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171 |
show "(#0::real) <= rinv (norm x)"; |
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|
172 |
by (rule real_less_imp_le, rule real_rinv_gt_zero1, |
7917 | 173 |
rule normed_vs_norm_gt_zero); |
8838 | 174 |
from a; show "abs (f x) <= c * norm x"; ..; |
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175 |
qed; |
7808 | 176 |
also; have "... = c * (norm x * rinv (norm x))"; |
177 |
by (rule real_mult_assoc); |
|
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178 |
also; have "(norm x * rinv (norm x)) = (#1::real)"; |
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179 |
proof (rule real_mult_inv_right1); |
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180 |
show nz: "norm x ~= (#0::real)"; |
7917 | 181 |
by (rule not_sym, rule lt_imp_not_eq, |
182 |
rule normed_vs_norm_gt_zero); |
|
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183 |
qed; |
7917 | 184 |
also; have "c * ... <= b "; by (simp! add: le_max1); |
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185 |
finally; show "y <= b"; .; |
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186 |
qed simp; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
187 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
188 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
189 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
190 |
|
7978 | 191 |
text {* The norm of a continuous function is always $\geq 0$. *}; |
7917 | 192 |
|
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193 |
lemma fnorm_ge_zero [intro??]: |
7978 | 194 |
"[| is_continuous V norm f; is_normed_vectorspace V norm|] |
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195 |
==> (#0::real) <= function_norm V norm f"; |
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196 |
proof -; |
7978 | 197 |
assume c: "is_continuous V norm f" |
7917 | 198 |
and n: "is_normed_vectorspace V norm"; |
199 |
||
200 |
txt {* The function norm is defined as the supremum of $B$. |
|
201 |
So it is $\geq 0$ if all elements in $B$ are $\geq 0$, provided |
|
202 |
the supremum exists and $B$ is not empty. *}; |
|
203 |
||
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204 |
show ?thesis; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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|
205 |
proof (unfold function_norm_def, rule sup_ub1); |
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206 |
show "ALL x:(B V norm f). (#0::real) <= x"; |
7978 | 207 |
proof (intro ballI, unfold B_def, |
208 |
elim UnE singletonE CollectE exE conjE); |
|
209 |
fix x r; |
|
8703 | 210 |
assume "x : V" "x ~= 00" |
8838 | 211 |
and r: "r = abs (f x) * rinv (norm x)"; |
7917 | 212 |
|
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213 |
have ge: "(#0::real) <= abs (f x)"; by (simp! only: abs_ge_zero); |
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parents:
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214 |
have "(#0::real) <= rinv (norm x)"; |
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215 |
by (rule real_less_imp_le, rule real_rinv_gt_zero1, rule);(*** |
7656 | 216 |
proof (rule real_less_imp_le); |
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217 |
show "(#0::real) < rinv (norm x)"; |
7566 | 218 |
proof (rule real_rinv_gt_zero); |
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219 |
show "(#0::real) < norm x"; ..; |
7566 | 220 |
qed; |
7917 | 221 |
qed; ***) |
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222 |
with ge; show "(#0::real) <= r"; |
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223 |
by (simp only: r, rule real_le_mult_order1a); |
7566 | 224 |
qed (simp!); |
7917 | 225 |
|
7978 | 226 |
txt {* Since $f$ is continuous the function norm exists: *}; |
7917 | 227 |
|
228 |
have "is_function_norm V norm f (function_norm V norm f)"; ..; |
|
229 |
thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
|
7978 | 230 |
by (unfold is_function_norm_def function_norm_def); |
7917 | 231 |
|
7978 | 232 |
txt {* $B$ is non-empty by construction: *}; |
7917 | 233 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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diff
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|
234 |
show "(#0::real) : B V norm f"; by (rule B_not_empty); |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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parents:
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|
235 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
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|
236 |
qed; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
237 |
|
7978 | 238 |
text{* \medskip The fundamental property of function norms is: |
7917 | 239 |
\begin{matharray}{l} |
240 |
| f\ap x | \leq {\fnorm {f}} \cdot {\norm x} |
|
241 |
\end{matharray} |
|
242 |
*}; |
|
243 |
||
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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diff
changeset
|
244 |
lemma norm_fx_le_norm_f_norm_x: |
7978 | 245 |
"[| is_normed_vectorspace V norm; x:V; is_continuous V norm f |] |
8838 | 246 |
==> abs (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
|
247 |
proof -; |
7917 | 248 |
assume "is_normed_vectorspace V norm" "x:V" |
7978 | 249 |
and c: "is_continuous V norm f"; |
7566 | 250 |
have v: "is_vectorspace V"; ..; |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
251 |
assume "x:V"; |
7917 | 252 |
|
253 |
txt{* The proof is by case analysis on $x$. *}; |
|
254 |
||
7927 | 255 |
show ?thesis; |
8280 | 256 |
proof cases; |
7917 | 257 |
|
258 |
txt {* For the case $x = \zero$ the thesis follows |
|
259 |
from the linearity of $f$: for every linear function |
|
260 |
holds $f\ap \zero = 0$. *}; |
|
261 |
||
8703 | 262 |
assume "x = 00"; |
8838 | 263 |
have "abs (f x) = abs (f 00)"; by (simp!); |
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
8838
diff
changeset
|
264 |
also; from v continuous_linearform; have "f 00 = (#0::real)"; ..; |
8838 | 265 |
also; note abs_zero; |
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
266 |
also; have "(#0::real) <= function_norm V norm f * norm x"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
267 |
proof (rule real_le_mult_order1a); |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
268 |
show "(#0::real) <= function_norm V norm f"; ..; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
269 |
show "(#0::real) <= norm x"; ..; |
7917 | 270 |
qed; |
271 |
finally; |
|
8838 | 272 |
show "abs (f x) <= function_norm V norm f * norm x"; .; |
7917 | 273 |
|
274 |
next; |
|
8703 | 275 |
assume "x ~= 00"; |
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
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diff
changeset
|
276 |
have n: "(#0::real) <= norm x"; ..; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
8838
diff
changeset
|
277 |
have nz: "norm x ~= (#0::real)"; |
7917 | 278 |
proof (rule lt_imp_not_eq [RS not_sym]); |
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
8838
diff
changeset
|
279 |
show "(#0::real) < norm x"; ..; |
7535
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
280 |
qed; |
7917 | 281 |
|
282 |
txt {* For the case $x\neq \zero$ we derive the following |
|
283 |
fact from the definition of the function norm:*}; |
|
284 |
||
8838 | 285 |
have l: "abs (f x) * rinv (norm x) <= function_norm V norm f"; |
7917 | 286 |
proof (unfold function_norm_def, rule sup_ub); |
287 |
from ex_fnorm [OF _ c]; |
|
288 |
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
|
289 |
by (simp! add: is_function_norm_def function_norm_def); |
|
8838 | 290 |
show "abs (f x) * rinv (norm x) : B V norm f"; |
7978 | 291 |
by (unfold B_def, intro UnI2 CollectI exI [of _ x] |
7917 | 292 |
conjI, simp); |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
293 |
qed; |
7917 | 294 |
|
7978 | 295 |
txt {* The thesis now follows by a short calculation: *}; |
7917 | 296 |
|
9013
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Updated files to remove 0r and 1r from theorems in descendant theories
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parents:
8838
diff
changeset
|
297 |
have "abs (f x) = abs (f x) * (#1::real)"; by (simp!); |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
298 |
also; from nz; have "(#1::real) = rinv (norm x) * norm x"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
299 |
by (rule real_mult_inv_left1 [RS sym]); |
7917 | 300 |
also; |
8838 | 301 |
have "abs (f x) * ... = abs (f x) * rinv (norm x) * norm x"; |
302 |
by (simp! add: real_mult_assoc [of "abs (f x)"]); |
|
7917 | 303 |
also; have "... <= function_norm V norm f * norm x"; |
304 |
by (rule real_mult_le_le_mono2 [OF n l]); |
|
305 |
finally; |
|
8838 | 306 |
show "abs (f x) <= function_norm V norm f * norm x"; .; |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
307 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
308 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
309 |
|
7978 | 310 |
text{* \medskip The function norm is the least positive real number for |
311 |
which the following inequation holds: |
|
7917 | 312 |
\begin{matharray}{l} |
313 |
| f\ap x | \leq c \cdot {\norm x} |
|
314 |
\end{matharray} |
|
315 |
*}; |
|
316 |
||
7535
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
317 |
lemma fnorm_le_ub: |
7978 | 318 |
"[| is_normed_vectorspace V norm; is_continuous V norm f; |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
319 |
ALL x:V. abs (f x) <= c * norm x; (#0::real) <= c |] |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
320 |
==> function_norm V norm f <= c"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
321 |
proof (unfold function_norm_def); |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
322 |
assume "is_normed_vectorspace V norm"; |
7978 | 323 |
assume c: "is_continuous V norm f"; |
8838 | 324 |
assume fb: "ALL x:V. abs (f x) <= c * norm x" |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
325 |
and "(#0::real) <= c"; |
7917 | 326 |
|
327 |
txt {* Suppose the inequation holds for some $c\geq 0$. |
|
328 |
If $c$ is an upper bound of $B$, then $c$ is greater |
|
329 |
than the function norm since the function norm is the |
|
330 |
least upper bound. |
|
331 |
*}; |
|
332 |
||
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
333 |
show "Sup UNIV (B V norm f) <= c"; |
7656 | 334 |
proof (rule sup_le_ub); |
7808 | 335 |
from ex_fnorm [OF _ c]; |
336 |
show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"; |
|
7566 | 337 |
by (simp! add: is_function_norm_def function_norm_def); |
7917 | 338 |
|
339 |
txt {* $c$ is an upper bound of $B$, i.~e.~every |
|
340 |
$y\in B$ is less than $c$. *}; |
|
341 |
||
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
342 |
show "isUb UNIV (B V norm f) c"; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
343 |
proof (intro isUbI setleI ballI); |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
344 |
fix y; assume "y: B V norm f"; |
7566 | 345 |
thus le: "y <= c"; |
7978 | 346 |
proof (unfold B_def, elim UnE CollectE exE conjE singletonE); |
7917 | 347 |
|
348 |
txt {* The first case for $y\in B$ is $y=0$. *}; |
|
349 |
||
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
350 |
assume "y = (#0::real)"; |
7917 | 351 |
show "y <= c"; by (force!); |
352 |
||
353 |
txt{* The second case is |
|
7978 | 354 |
$y = {|f\ap x|}/{\norm x}$ for some |
7917 | 355 |
$x\in V$ with $x \neq \zero$. *}; |
356 |
||
357 |
next; |
|
358 |
fix x; |
|
8703 | 359 |
assume "x : V" "x ~= 00"; |
7917 | 360 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
361 |
have lz: "(#0::real) < norm x"; |
7917 | 362 |
by (simp! add: normed_vs_norm_gt_zero); |
7566 | 363 |
|
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
364 |
have nz: "norm x ~= (#0::real)"; |
7566 | 365 |
proof (rule not_sym); |
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
366 |
from lz; show "(#0::real) ~= norm x"; |
7917 | 367 |
by (simp! add: order_less_imp_not_eq); |
7566 | 368 |
qed; |
369 |
||
9013
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
370 |
from lz; have "(#0::real) < rinv (norm x)"; |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
371 |
by (simp! add: real_rinv_gt_zero1); |
9dd0274f76af
Updated files to remove 0r and 1r from theorems in descendant theories
fleuriot
parents:
8838
diff
changeset
|
372 |
hence rinv_gez: "(#0::real) <= rinv (norm x)"; |
7808 | 373 |
by (rule real_less_imp_le); |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
374 |
|
8838 | 375 |
assume "y = abs (f x) * rinv (norm x)"; |
7917 | 376 |
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
|
377 |
proof (rule real_mult_le_le_mono2); |
8838 | 378 |
show "abs (f x) <= c * norm x"; by (rule bspec); |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
379 |
qed; |
7917 | 380 |
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
|
381 |
finally; show ?thesis; .; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
382 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
383 |
qed force; |
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 |
qed; |
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
diff
changeset
|
386 |
|
7808 | 387 |
end; |