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src/HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy

author | wenzelm |

Sat Nov 04 15:24:40 2017 +0100 (20 months ago) | |

changeset 67003 | 49850a679c2c |

parent 63040 | eb4ddd18d635 |

permissions | -rw-r--r-- |

more robust sorted_entries;

1 (* Title: HOL/Hahn_Banach/Hahn_Banach_Ext_Lemmas.thy

2 Author: Gertrud Bauer, TU Munich

3 *)

5 section \<open>Extending non-maximal functions\<close>

7 theory Hahn_Banach_Ext_Lemmas

8 imports Function_Norm

9 begin

11 text \<open>

12 In this section the following context is presumed. Let \<open>E\<close> be a real vector

13 space with a seminorm \<open>q\<close> on \<open>E\<close>. \<open>F\<close> is a subspace of \<open>E\<close> and \<open>f\<close> a linear

14 function on \<open>F\<close>. We consider a subspace \<open>H\<close> of \<open>E\<close> that is a superspace of

15 \<open>F\<close> and a linear form \<open>h\<close> on \<open>H\<close>. \<open>H\<close> is a not equal to \<open>E\<close> and \<open>x\<^sub>0\<close> is an

16 element in \<open>E - H\<close>. \<open>H\<close> is extended to the direct sum \<open>H' = H + lin x\<^sub>0\<close>, so

17 for any \<open>x \<in> H'\<close> the decomposition of \<open>x = y + a \<cdot> x\<close> with \<open>y \<in> H\<close> is

18 unique. \<open>h'\<close> is defined on \<open>H'\<close> by \<open>h' x = h y + a \<cdot> \<xi>\<close> for a certain \<open>\<xi>\<close>.

20 Subsequently we show some properties of this extension \<open>h'\<close> of \<open>h\<close>.

22 \<^medskip>

23 This lemma will be used to show the existence of a linear extension of \<open>f\<close>

24 (see page \pageref{ex-xi-use}). It is a consequence of the completeness of

25 \<open>\<real>\<close>. To show

26 \begin{center}

27 \begin{tabular}{l}

28 \<open>\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y\<close>

29 \end{tabular}

30 \end{center}

31 \<^noindent> it suffices to show that

32 \begin{center}

33 \begin{tabular}{l}

34 \<open>\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v\<close>

35 \end{tabular}

36 \end{center}

37 \<close>

39 lemma ex_xi:

40 assumes "vectorspace F"

41 assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"

42 shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"

43 proof -

44 interpret vectorspace F by fact

45 txt \<open>From the completeness of the reals follows:

46 The set \<open>S = {a u. u \<in> F}\<close> has a supremum, if it is

47 non-empty and has an upper bound.\<close>

49 let ?S = "{a u | u. u \<in> F}"

50 have "\<exists>xi. lub ?S xi"

51 proof (rule real_complete)

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

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

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

55 proof

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

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

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

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

60 qed

61 then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..

62 qed

63 then obtain xi where xi: "lub ?S xi" ..

64 {

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

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

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

68 }

69 moreover {

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

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

72 proof (rule lub.least)

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

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

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

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

77 qed

78 }

79 ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast

80 qed

82 text \<open>

83 \<^medskip>

84 The function \<open>h'\<close> is defined as a \<open>h' x = h y + a \<cdot> \<xi>\<close> where \<open>x = y + a \<cdot> \<xi>\<close>

85 is a linear extension of \<open>h\<close> to \<open>H'\<close>.

86 \<close>

88 lemma h'_lf:

89 assumes h'_def: "\<And>x. h' x = (let (y, a) =

90 SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"

91 and H'_def: "H' = H + lin x0"

92 and HE: "H \<unlhd> E"

93 assumes "linearform H h"

94 assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"

95 assumes E: "vectorspace E"

96 shows "linearform H' h'"

97 proof -

98 interpret linearform H h by fact

99 interpret vectorspace E by fact

100 show ?thesis

101 proof

102 note E = \<open>vectorspace E\<close>

103 have H': "vectorspace H'"

104 proof (unfold H'_def)

105 from \<open>x0 \<in> E\<close>

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

107 with HE show "vectorspace (H + lin x0)" using E ..

108 qed

109 {

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

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

112 proof -

113 from H' x1 x2 have "x1 + x2 \<in> H'"

114 by (rule vectorspace.add_closed)

115 with x1 x2 obtain y y1 y2 a a1 a2 where

116 x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"

117 and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

118 and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"

119 unfolding H'_def sum_def lin_def by blast

121 have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0

122 proof (rule decomp_H') text_raw \<open>\label{decomp-H-use}\<close>

123 from HE y1 y2 show "y1 + y2 \<in> H"

124 by (rule subspace.add_closed)

125 from x0 and HE y y1 y2

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

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

128 by (simp add: add_ac add_mult_distrib2)

129 also note x1x2

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

131 qed

133 from h'_def x1x2 E HE y x0

134 have "h' (x1 + x2) = h y + a * xi"

135 by (rule h'_definite)

136 also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"

137 by (simp only: ya)

138 also from y1 y2 have "h (y1 + y2) = h y1 + h y2"

139 by simp

140 also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"

141 by (simp add: distrib_right)

142 also from h'_def x1_rep E HE y1 x0

143 have "h y1 + a1 * xi = h' x1"

144 by (rule h'_definite [symmetric])

145 also from h'_def x2_rep E HE y2 x0

146 have "h y2 + a2 * xi = h' x2"

147 by (rule h'_definite [symmetric])

148 finally show ?thesis .

149 qed

150 next

151 fix x1 c assume x1: "x1 \<in> H'"

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

153 proof -

154 from H' x1 have ax1: "c \<cdot> x1 \<in> H'"

155 by (rule vectorspace.mult_closed)

156 with x1 obtain y a y1 a1 where

157 cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"

158 and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"

159 unfolding H'_def sum_def lin_def by blast

161 have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using E HE _ y x0

162 proof (rule decomp_H')

163 from HE y1 show "c \<cdot> y1 \<in> H"

164 by (rule subspace.mult_closed)

165 from x0 and HE y y1

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

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

168 by (simp add: mult_assoc add_mult_distrib1)

169 also note cx1_rep

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

171 qed

173 from h'_def cx1_rep E HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"

174 by (rule h'_definite)

175 also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"

176 by (simp only: ya)

177 also from y1 have "h (c \<cdot> y1) = c * h y1"

178 by simp

179 also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"

180 by (simp only: distrib_left)

181 also from h'_def x1_rep E HE y1 x0 have "h y1 + a1 * xi = h' x1"

182 by (rule h'_definite [symmetric])

183 finally show ?thesis .

184 qed

185 }

186 qed

187 qed

189 text \<open>

190 \<^medskip>

191 The linear extension \<open>h'\<close> of \<open>h\<close> is bounded by the seminorm \<open>p\<close>.

192 \<close>

194 lemma h'_norm_pres:

195 assumes h'_def: "\<And>x. h' x = (let (y, a) =

196 SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi)"

197 and H'_def: "H' = H + lin x0"

198 and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"

199 assumes E: "vectorspace E" and HE: "subspace H E"

200 and "seminorm E p" and "linearform H h"

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"

204 proof -

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

209 show ?thesis

210 proof

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"

215 and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto

216 from x' obtain y a where

217 x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"

218 unfolding H'_def sum_def lin_def by blast

219 from y have y': "y \<in> E" ..

220 from y have ay: "inverse a \<cdot> y \<in> H" by simp

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

223 by (rule h'_definite)

224 also have "\<dots> \<le> p (y + a \<cdot> x0)"

225 proof (rule linorder_cases)

226 assume z: "a = 0"

227 then have "h y + a * xi = h y" by simp

228 also from a y have "\<dots> \<le> p y" ..

229 also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp

230 finally show ?thesis .

231 next

232 txt \<open>In the case \<open>a < 0\<close>, we use \<open>a\<^sub>1\<close>

233 with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>

234 assume lz: "a < 0" then have nz: "a \<noteq> 0" by simp

235 from a1 ay

236 have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..

237 with lz have "a * xi \<le>

238 a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

239 by (simp add: mult_left_mono_neg order_less_imp_le)

241 also have "\<dots> =

242 - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"

243 by (simp add: right_diff_distrib)

244 also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =

245 p (a \<cdot> (inverse a \<cdot> y + x0))"

246 by (simp add: abs_homogenous)

247 also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

248 by (simp add: add_mult_distrib1 mult_assoc [symmetric])

249 also from nz y have "a * (h (inverse a \<cdot> y)) = h y"

250 by simp

251 finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

252 then show ?thesis by simp

253 next

254 txt \<open>In the case \<open>a > 0\<close>, we use \<open>a\<^sub>2\<close>

255 with \<open>ya\<close> taken as \<open>y / a\<close>:\<close>

256 assume gz: "0 < a" then have nz: "a \<noteq> 0" by simp

257 from a2 ay

258 have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..

259 with gz have "a * xi \<le>

260 a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"

261 by simp

262 also have "\<dots> = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"

263 by (simp add: right_diff_distrib)

264 also from gz x0 y'

265 have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"

266 by (simp add: abs_homogenous)

267 also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"

268 by (simp add: add_mult_distrib1 mult_assoc [symmetric])

269 also from nz y have "a * h (inverse a \<cdot> y) = h y"

270 by simp

271 finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .

272 then show ?thesis by simp

273 qed

274 also from x_rep have "\<dots> = p x" by (simp only:)

275 finally show ?thesis .

276 qed

277 qed

278 qed

280 end