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(* Title: HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy 
7917  2 
Author: Gertrud Bauer, TU Munich 
3 
*) 

4 

10007  5 
header {* Extending nonmaximal functions *} 
7917  6 

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theory Hahn_Banach_Ext_Lemmas 
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imports Function_Norm 
27612  9 
begin 
7917  10 

10687  11 
text {* 
12 
In this section the following context is presumed. Let @{text E} be 

13 
a real vector space with a seminorm @{text q} on @{text E}. @{text 

14 
F} is a subspace of @{text E} and @{text f} a linear function on 

15 
@{text F}. We consider a subspace @{text H} of @{text E} that is a 

16 
superspace of @{text F} and a linear form @{text h} on @{text 

17 
H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is 

18 
an element in @{text "E  H"}. @{text H} is extended to the direct 

19 
sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"} 

20 
the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is 

13515  21 
unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y + 
22 
a \<cdot> \<xi>"} for a certain @{text \<xi>}. 

7917  23 

10687  24 
Subsequently we show some properties of this extension @{text h'} of 
25 
@{text h}. 

7917  26 

13515  27 
\medskip This lemma will be used to show the existence of a linear 
28 
extension of @{text f} (see page \pageref{exxiuse}). It is a 

29 
consequence of the completeness of @{text \<real>}. To show 

10687  30 
\begin{center} 
31 
\begin{tabular}{l} 

32 
@{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"} 

33 
\end{tabular} 

34 
\end{center} 

35 
\noindent it suffices to show that 

36 
\begin{center} 

37 
\begin{tabular}{l} 

38 
@{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"} 

39 
\end{tabular} 

40 
\end{center} 

41 
*} 

7917  42 

10687  43 
lemma ex_xi: 
27611  44 
assumes "vectorspace F" 
13515  45 
assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v" 
46 
shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" 

10007  47 
proof  
29234  48 
interpret vectorspace F by fact 
7917  49 
txt {* From the completeness of the reals follows: 
13515  50 
The set @{text "S = {a u. u \<in> F}"} has a supremum, if it is 
51 
nonempty and has an upper bound. *} 

7917  52 

13515  53 
let ?S = "{a u  u. u \<in> F}" 
54 
have "\<exists>xi. lub ?S xi" 

55 
proof (rule real_complete) 

56 
have "a 0 \<in> ?S" by blast 

57 
then show "\<exists>X. X \<in> ?S" .. 

58 
have "\<forall>y \<in> ?S. y \<le> b 0" 

59 
proof 

60 
fix y assume y: "y \<in> ?S" 

61 
then obtain u where u: "u \<in> F" and y: "y = a u" by blast 

62 
from u and zero have "a u \<le> b 0" by (rule r) 

63 
with y show "y \<le> b 0" by (simp only:) 

10007  64 
qed 
13515  65 
then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" .. 
10007  66 
qed 
13515  67 
then obtain xi where xi: "lub ?S xi" .. 
68 
{ 

69 
fix y assume "y \<in> F" 

70 
then have "a y \<in> ?S" by blast 

71 
with xi have "a y \<le> xi" by (rule lub.upper) 

72 
} moreover { 

73 
fix y assume y: "y \<in> F" 

74 
from xi have "xi \<le> b y" 

75 
proof (rule lub.least) 

76 
fix au assume "au \<in> ?S" 

77 
then obtain u where u: "u \<in> F" and au: "au = a u" by blast 

78 
from u y have "a u \<le> b y" by (rule r) 

79 
with au show "au \<le> b y" by (simp only:) 

10007  80 
qed 
13515  81 
} ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast 
10007  82 
qed 
7917  83 

10687  84 
text {* 
13515  85 
\medskip The function @{text h'} is defined as a @{text "h' x = h y 
86 
+ a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of 

87 
@{text h} to @{text H'}. 

88 
*} 

7917  89 

10687  90 
lemma h'_lf: 
13515  91 
assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) = 
92 
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi" 

44190  93 
and H'_def: "H' \<equiv> H \<oplus> lin x0" 
13515  94 
and HE: "H \<unlhd> E" 
27611  95 
assumes "linearform H h" 
13515  96 
assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" 
27611  97 
assumes E: "vectorspace E" 
13515  98 
shows "linearform H' h'" 
27611  99 
proof  
29234  100 
interpret linearform H h by fact 
101 
interpret vectorspace E by fact 

27612  102 
show ?thesis 
103 
proof 

27611  104 
note E = `vectorspace E` 
105 
have H': "vectorspace H'" 

106 
proof (unfold H'_def) 

107 
from `x0 \<in> E` 

108 
have "lin x0 \<unlhd> E" .. 

44190  109 
with HE show "vectorspace (H \<oplus> lin x0)" using E .. 
27611  110 
qed 
111 
{ 

112 
fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'" 

113 
show "h' (x1 + x2) = h' x1 + h' x2" 

114 
proof  

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from H' x1 x2 have "x1 + x2 \<in> H'" 
27611  116 
by (rule vectorspace.add_closed) 
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with x1 x2 obtain y y1 y2 a a1 a2 where 
27611  118 
x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H" 
13515  119 
and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H" 
120 
and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H" 

27612  121 
unfolding H'_def sum_def lin_def by blast 
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have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0 
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proof (rule decomp_H') txt_raw {* \label{decompHuse} *} 
27611  125 
from HE y1 y2 show "y1 + y2 \<in> H" 
126 
by (rule subspace.add_closed) 

127 
from x0 and HE y y1 y2 

128 
have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto 

129 
with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2" 

130 
by (simp add: add_ac add_mult_distrib2) 

131 
also note x1x2 

132 
finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" . 

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qed 
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from h'_def x1x2 E HE y x0 
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have "h' (x1 + x2) = h y + a * xi" 
27611  137 
by (rule h'_definite) 
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also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi" 
27611  139 
by (simp only: ya) 
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also from y1 y2 have "h (y1 + y2) = h y1 + h y2" 
27611  141 
by simp 
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also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)" 
27611  143 
by (simp add: left_distrib) 
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also from h'_def x1_rep E HE y1 x0 
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have "h y1 + a1 * xi = h' x1" 
27611  146 
by (rule h'_definite [symmetric]) 
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also from h'_def x2_rep E HE y2 x0 
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have "h y2 + a2 * xi = h' x2" 
27611  149 
by (rule h'_definite [symmetric]) 
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finally show ?thesis . 
10007  151 
qed 
27611  152 
next 
153 
fix x1 c assume x1: "x1 \<in> H'" 

154 
show "h' (c \<cdot> x1) = c * (h' x1)" 

155 
proof  

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from H' x1 have ax1: "c \<cdot> x1 \<in> H'" 
27611  157 
by (rule vectorspace.mult_closed) 
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with x1 obtain y a y1 a1 where 
27612  159 
cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H" 
13515  160 
and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H" 
27612  161 
unfolding H'_def sum_def lin_def by blast 
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have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0 
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proof (rule decomp_H') 
27611  165 
from HE y1 show "c \<cdot> y1 \<in> H" 
166 
by (rule subspace.mult_closed) 

167 
from x0 and HE y y1 

168 
have "x0 \<in> E" "y \<in> E" "y1 \<in> E" by auto 

169 
with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1" 

170 
by (simp add: mult_assoc add_mult_distrib1) 

171 
also note cx1_rep 

172 
finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" . 

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qed 
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from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi" 
27611  176 
by (rule h'_definite) 
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also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi" 
27611  178 
by (simp only: ya) 
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also from y1 have "h (c \<cdot> y1) = c * h y1" 
27611  180 
by simp 
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also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)" 
27611  182 
by (simp only: right_distrib) 
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also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1" 
27611  184 
by (rule h'_definite [symmetric]) 
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finally show ?thesis . 
10007  186 
qed 
27611  187 
} 
188 
qed 

10007  189 
qed 
7917  190 

10687  191 
text {* \medskip The linear extension @{text h'} of @{text h} 
13515  192 
is bounded by the seminorm @{text p}. *} 
7917  193 

9374  194 
lemma h'_norm_pres: 
13515  195 
assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) = 
196 
SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi" 

44190  197 
and H'_def: "H' \<equiv> H \<oplus> lin x0" 
13515  198 
and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" 
27611  199 
assumes E: "vectorspace E" and HE: "subspace H E" 
200 
and "seminorm E p" and "linearform H h" 

13515  201 
assumes a: "\<forall>y \<in> H. h y \<le> p y" 
202 
and a': "\<forall>y \<in> H.  p (y + x0)  h y \<le> xi \<and> xi \<le> p (y + x0)  h y" 

203 
shows "\<forall>x \<in> H'. h' x \<le> p x" 

27611  204 
proof  
29234  205 
interpret vectorspace E by fact 
206 
interpret subspace H E by fact 

207 
interpret seminorm E p by fact 

208 
interpret linearform H h by fact 

27612  209 
show ?thesis 
210 
proof 

27611  211 
fix x assume x': "x \<in> H'" 
212 
show "h' x \<le> p x" 

213 
proof  

214 
from a' have a1: "\<forall>ya \<in> H.  p (ya + x0)  h ya \<le> xi" 

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and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0)  h ya" by auto 
27611  216 
from x' obtain y a where 
27612  217 
x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H" 
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unfolding H'_def sum_def lin_def by blast 
27611  219 
from y have y': "y \<in> E" .. 
220 
from y have ay: "inverse a \<cdot> y \<in> H" by simp 

221 

222 
from h'_def x_rep E HE y x0 have "h' x = h y + a * xi" 

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by (rule h'_definite) 
27611  224 
also have "\<dots> \<le> p (y + a \<cdot> x0)" 
225 
proof (rule linorder_cases) 

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assume z: "a = 0" 
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then have "h y + a * xi = h y" by simp 
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also from a y have "\<dots> \<le> p y" .. 
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also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp 
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finally show ?thesis . 
27611  231 
next 
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txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"} 
27611  233 
with @{text ya} taken as @{text "y / a"}: *} 
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assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp 
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from a1 ay 
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have " p (inverse a \<cdot> y + x0)  h (inverse a \<cdot> y) \<le> xi" .. 
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with lz have "a * xi \<le> 
13515  238 
a * ( p (inverse a \<cdot> y + x0)  h (inverse a \<cdot> y))" 
27611  239 
by (simp add: mult_left_mono_neg order_less_imp_le) 
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also have "\<dots> = 
13515  242 
 a * (p (inverse a \<cdot> y + x0))  a * (h (inverse a \<cdot> y))" 
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by (simp add: right_diff_distrib) 
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also from lz x0 y' have " a * (p (inverse a \<cdot> y + x0)) = 
13515  245 
p (a \<cdot> (inverse a \<cdot> y + x0))" 
27611  246 
by (simp add: abs_homogenous) 
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also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)" 
27611  248 
by (simp add: add_mult_distrib1 mult_assoc [symmetric]) 
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also from nz y have "a * (h (inverse a \<cdot> y)) = h y" 
27611  250 
by simp 
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finally have "a * xi \<le> p (y + a \<cdot> x0)  h y" . 
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then show ?thesis by simp 
27611  253 
next 
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txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"} 
27611  255 
with @{text ya} taken as @{text "y / a"}: *} 
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assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp 
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from a2 ay 
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have "xi \<le> p (inverse a \<cdot> y + x0)  h (inverse a \<cdot> y)" .. 
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with gz have "a * xi \<le> 
13515  260 
a * (p (inverse a \<cdot> y + x0)  h (inverse a \<cdot> y))" 
27611  261 
by simp 
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diff
changeset

262 
also have "\<dots> = a * p (inverse a \<cdot> y + x0)  a * h (inverse a \<cdot> y)" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31795
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changeset

263 
by (simp add: right_diff_distrib) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31795
diff
changeset

264 
also from gz x0 y' 
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eliminated hard tabulators, guessing at each author's individual tabwidth;
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changeset

265 
have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))" 
27611  266 
by (simp add: abs_homogenous) 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31795
diff
changeset

267 
also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)" 
27611  268 
by (simp add: add_mult_distrib1 mult_assoc [symmetric]) 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31795
diff
changeset

269 
also from nz y have "a * h (inverse a \<cdot> y) = h y" 
27611  270 
by simp 
32960
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parents:
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changeset

271 
finally have "a * xi \<le> p (y + a \<cdot> x0)  h y" . 
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272 
then show ?thesis by simp 
27611  273 
qed 
274 
also from x_rep have "\<dots> = p x" by (simp only:) 

275 
finally show ?thesis . 

10007  276 
qed 
277 
qed 

13515  278 
qed 
7917  279 

10007  280 
end 