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src/HOL/Hahn_Banach/Hahn_Banach.thy

author | wenzelm |

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

changeset 67003 | 49850a679c2c |

parent 65166 | f8aafbf2b02e |

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

more robust sorted_entries;

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

2 Author: Gertrud Bauer, TU Munich

3 *)

5 section \<open>The Hahn-Banach Theorem\<close>

7 theory Hahn_Banach

8 imports Hahn_Banach_Lemmas

9 begin

11 text \<open>

12 We present the proof of two different versions of the Hahn-Banach Theorem,

13 closely following @{cite \<open>\S36\<close> "Heuser:1986"}.

14 \<close>

17 subsection \<open>The Hahn-Banach Theorem for vector spaces\<close>

19 paragraph \<open>Hahn-Banach Theorem.\<close>

20 text \<open>

21 Let \<open>F\<close> be a subspace of a real vector space \<open>E\<close>, let \<open>p\<close> be a semi-norm on

22 \<open>E\<close>, and \<open>f\<close> be a linear form defined on \<open>F\<close> such that \<open>f\<close> is bounded by

23 \<open>p\<close>, i.e. \<open>\<forall>x \<in> F. f x \<le> p x\<close>. Then \<open>f\<close> can be extended to a linear form \<open>h\<close>

24 on \<open>E\<close> such that \<open>h\<close> is norm-preserving, i.e. \<open>h\<close> is also bounded by \<open>p\<close>.

25 \<close>

27 paragraph \<open>Proof Sketch.\<close>

28 text \<open>

29 \<^enum> Define \<open>M\<close> as the set of norm-preserving extensions of \<open>f\<close> to subspaces of

30 \<open>E\<close>. The linear forms in \<open>M\<close> are ordered by domain extension.

32 \<^enum> We show that every non-empty chain in \<open>M\<close> has an upper bound in \<open>M\<close>.

34 \<^enum> With Zorn's Lemma we conclude that there is a maximal function \<open>g\<close> in \<open>M\<close>.

36 \<^enum> The domain \<open>H\<close> of \<open>g\<close> is the whole space \<open>E\<close>, as shown by classical

37 contradiction:

39 \<^item> Assuming \<open>g\<close> is not defined on whole \<open>E\<close>, it can still be extended in a

40 norm-preserving way to a super-space \<open>H'\<close> of \<open>H\<close>.

42 \<^item> Thus \<open>g\<close> can not be maximal. Contradiction!

43 \<close>

45 theorem Hahn_Banach:

46 assumes E: "vectorspace E" and "subspace F E"

47 and "seminorm E p" and "linearform F f"

48 assumes fp: "\<forall>x \<in> F. f x \<le> p x"

49 shows "\<exists>h. linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)"

50 \<comment> \<open>Let \<open>E\<close> be a vector space, \<open>F\<close> a subspace of \<open>E\<close>, \<open>p\<close> a seminorm on \<open>E\<close>,\<close>

51 \<comment> \<open>and \<open>f\<close> a linear form on \<open>F\<close> such that \<open>f\<close> is bounded by \<open>p\<close>,\<close>

52 \<comment> \<open>then \<open>f\<close> can be extended to a linear form \<open>h\<close> on \<open>E\<close> in a norm-preserving way. \<^smallskip>\<close>

53 proof -

54 interpret vectorspace E by fact

55 interpret subspace F E by fact

56 interpret seminorm E p by fact

57 interpret linearform F f by fact

58 define M where "M = norm_pres_extensions E p F f"

59 then have M: "M = \<dots>" by (simp only:)

60 from E have F: "vectorspace F" ..

61 note FE = \<open>F \<unlhd> E\<close>

62 {

63 fix c assume cM: "c \<in> chains M" and ex: "\<exists>x. x \<in> c"

64 have "\<Union>c \<in> M"

65 \<comment> \<open>Show that every non-empty chain \<open>c\<close> of \<open>M\<close> has an upper bound in \<open>M\<close>:\<close>

66 \<comment> \<open>\<open>\<Union>c\<close> is greater than any element of the chain \<open>c\<close>, so it suffices to show \<open>\<Union>c \<in> M\<close>.\<close>

67 unfolding M_def

68 proof (rule norm_pres_extensionI)

69 let ?H = "domain (\<Union>c)"

70 let ?h = "funct (\<Union>c)"

72 have a: "graph ?H ?h = \<Union>c"

73 proof (rule graph_domain_funct)

74 fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"

75 with M_def cM show "z = y" by (rule sup_definite)

76 qed

77 moreover from M cM a have "linearform ?H ?h"

78 by (rule sup_lf)

79 moreover from a M cM ex FE E have "?H \<unlhd> E"

80 by (rule sup_subE)

81 moreover from a M cM ex FE have "F \<unlhd> ?H"

82 by (rule sup_supF)

83 moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"

84 by (rule sup_ext)

85 moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"

86 by (rule sup_norm_pres)

87 ultimately show "\<exists>H h. \<Union>c = graph H h

88 \<and> linearform H h

89 \<and> H \<unlhd> E

90 \<and> F \<unlhd> H

91 \<and> graph F f \<subseteq> graph H h

92 \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast

93 qed

94 }

95 then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"

96 \<comment> \<open>With Zorn's Lemma we can conclude that there is a maximal element in \<open>M\<close>. \<^smallskip>\<close>

97 proof (rule Zorn's_Lemma)

98 \<comment> \<open>We show that \<open>M\<close> is non-empty:\<close>

99 show "graph F f \<in> M"

100 unfolding M_def

101 proof (rule norm_pres_extensionI2)

102 show "linearform F f" by fact

103 show "F \<unlhd> E" by fact

104 from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)

105 show "graph F f \<subseteq> graph F f" ..

106 show "\<forall>x\<in>F. f x \<le> p x" by fact

107 qed

108 qed

109 then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"

110 by blast

111 from gM obtain H h where

112 g_rep: "g = graph H h"

113 and linearform: "linearform H h"

114 and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"

115 and graphs: "graph F f \<subseteq> graph H h"

116 and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..

117 \<comment> \<open>\<open>g\<close> is a norm-preserving extension of \<open>f\<close>, in other words:\<close>

118 \<comment> \<open>\<open>g\<close> is the graph of some linear form \<open>h\<close> defined on a subspace \<open>H\<close> of \<open>E\<close>,\<close>

119 \<comment> \<open>and \<open>h\<close> is an extension of \<open>f\<close> that is again bounded by \<open>p\<close>. \<^smallskip>\<close>

120 from HE E have H: "vectorspace H"

121 by (rule subspace.vectorspace)

123 have HE_eq: "H = E"

124 \<comment> \<open>We show that \<open>h\<close> is defined on whole \<open>E\<close> by classical contradiction. \<^smallskip>\<close>

125 proof (rule classical)

126 assume neq: "H \<noteq> E"

127 \<comment> \<open>Assume \<open>h\<close> is not defined on whole \<open>E\<close>. Then show that \<open>h\<close> can be extended\<close>

128 \<comment> \<open>in a norm-preserving way to a function \<open>h'\<close> with the graph \<open>g'\<close>. \<^smallskip>\<close>

129 have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"

130 proof -

131 from HE have "H \<subseteq> E" ..

132 with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast

133 obtain x': "x' \<noteq> 0"

134 proof

135 show "x' \<noteq> 0"

136 proof

137 assume "x' = 0"

138 with H have "x' \<in> H" by (simp only: vectorspace.zero)

139 with \<open>x' \<notin> H\<close> show False by contradiction

140 qed

141 qed

143 define H' where "H' = H + lin x'"

144 \<comment> \<open>Define \<open>H'\<close> as the direct sum of \<open>H\<close> and the linear closure of \<open>x'\<close>. \<^smallskip>\<close>

145 have HH': "H \<unlhd> H'"

146 proof (unfold H'_def)

147 from x'E have "vectorspace (lin x')" ..

148 with H show "H \<unlhd> H + lin x'" ..

149 qed

151 obtain xi where

152 xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi

153 \<and> xi \<le> p (y + x') - h y"

154 \<comment> \<open>Pick a real number \<open>\<xi>\<close> that fulfills certain inequality; this will\<close>

155 \<comment> \<open>be used to establish that \<open>h'\<close> is a norm-preserving extension of \<open>h\<close>.

156 \label{ex-xi-use}\<^smallskip>\<close>

157 proof -

158 from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi

159 \<and> xi \<le> p (y + x') - h y"

160 proof (rule ex_xi)

161 fix u v assume u: "u \<in> H" and v: "v \<in> H"

162 with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto

163 from H u v linearform have "h v - h u = h (v - u)"

164 by (simp add: linearform.diff)

165 also from hp and H u v have "\<dots> \<le> p (v - u)"

166 by (simp only: vectorspace.diff_closed)

167 also from x'E uE vE have "v - u = x' + - x' + v + - u"

168 by (simp add: diff_eq1)

169 also from x'E uE vE have "\<dots> = v + x' + - (u + x')"

170 by (simp add: add_ac)

171 also from x'E uE vE have "\<dots> = (v + x') - (u + x')"

172 by (simp add: diff_eq1)

173 also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"

174 by (simp add: diff_subadditive)

175 finally have "h v - h u \<le> p (v + x') + p (u + x')" .

176 then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp

177 qed

178 then show thesis by (blast intro: that)

179 qed

181 define h' where "h' x = (let (y, a) =

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

183 \<comment> \<open>Define the extension \<open>h'\<close> of \<open>h\<close> to \<open>H'\<close> using \<open>\<xi>\<close>. \<^smallskip>\<close>

185 have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"

186 \<comment> \<open>\<open>h'\<close> is an extension of \<open>h\<close> \dots \<^smallskip>\<close>

187 proof

188 show "g \<subseteq> graph H' h'"

189 proof -

190 have "graph H h \<subseteq> graph H' h'"

191 proof (rule graph_extI)

192 fix t assume t: "t \<in> H"

193 from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"

194 using \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> by (rule decomp_H'_H)

195 with h'_def show "h t = h' t" by (simp add: Let_def)

196 next

197 from HH' show "H \<subseteq> H'" ..

198 qed

199 with g_rep show ?thesis by (simp only:)

200 qed

202 show "g \<noteq> graph H' h'"

203 proof -

204 have "graph H h \<noteq> graph H' h'"

205 proof

206 assume eq: "graph H h = graph H' h'"

207 have "x' \<in> H'"

208 unfolding H'_def

209 proof

210 from H show "0 \<in> H" by (rule vectorspace.zero)

211 from x'E show "x' \<in> lin x'" by (rule x_lin_x)

212 from x'E show "x' = 0 + x'" by simp

213 qed

214 then have "(x', h' x') \<in> graph H' h'" ..

215 with eq have "(x', h' x') \<in> graph H h" by (simp only:)

216 then have "x' \<in> H" ..

217 with \<open>x' \<notin> H\<close> show False by contradiction

218 qed

219 with g_rep show ?thesis by simp

220 qed

221 qed

222 moreover have "graph H' h' \<in> M"

223 \<comment> \<open>and \<open>h'\<close> is norm-preserving. \<^smallskip>\<close>

224 proof (unfold M_def)

225 show "graph H' h' \<in> norm_pres_extensions E p F f"

226 proof (rule norm_pres_extensionI2)

227 show "linearform H' h'"

228 using h'_def H'_def HE linearform \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E

229 by (rule h'_lf)

230 show "H' \<unlhd> E"

231 unfolding H'_def

232 proof

233 show "H \<unlhd> E" by fact

234 show "vectorspace E" by fact

235 from x'E show "lin x' \<unlhd> E" ..

236 qed

237 from H \<open>F \<unlhd> H\<close> HH' show FH': "F \<unlhd> H'"

238 by (rule vectorspace.subspace_trans)

239 show "graph F f \<subseteq> graph H' h'"

240 proof (rule graph_extI)

241 fix x assume x: "x \<in> F"

242 with graphs have "f x = h x" ..

243 also have "\<dots> = h x + 0 * xi" by simp

244 also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"

245 by (simp add: Let_def)

246 also have "(x, 0) =

247 (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"

248 using E HE

249 proof (rule decomp_H'_H [symmetric])

250 from FH x show "x \<in> H" ..

251 from x' show "x' \<noteq> 0" .

252 show "x' \<notin> H" by fact

253 show "x' \<in> E" by fact

254 qed

255 also have

256 "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)

257 in h y + a * xi) = h' x" by (simp only: h'_def)

258 finally show "f x = h' x" .

259 next

260 from FH' show "F \<subseteq> H'" ..

261 qed

262 show "\<forall>x \<in> H'. h' x \<le> p x"

263 using h'_def H'_def \<open>x' \<notin> H\<close> \<open>x' \<in> E\<close> \<open>x' \<noteq> 0\<close> E HE

264 \<open>seminorm E p\<close> linearform and hp xi

265 by (rule h'_norm_pres)

266 qed

267 qed

268 ultimately show ?thesis ..

269 qed

270 then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp

271 \<comment> \<open>So the graph \<open>g\<close> of \<open>h\<close> cannot be maximal. Contradiction! \<^smallskip>\<close>

272 with gx show "H = E" by contradiction

273 qed

275 from HE_eq and linearform have "linearform E h"

276 by (simp only:)

277 moreover have "\<forall>x \<in> F. h x = f x"

278 proof

279 fix x assume "x \<in> F"

280 with graphs have "f x = h x" ..

281 then show "h x = f x" ..

282 qed

283 moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"

284 by (simp only:)

285 ultimately show ?thesis by blast

286 qed

289 subsection \<open>Alternative formulation\<close>

291 text \<open>

292 The following alternative formulation of the Hahn-Banach

293 Theorem\label{abs-Hahn-Banach} uses the fact that for a real linear form \<open>f\<close>

294 and a seminorm \<open>p\<close> the following inequality are equivalent:\footnote{This

295 was shown in lemma @{thm [source] abs_ineq_iff} (see page

296 \pageref{abs-ineq-iff}).}

297 \begin{center}

298 \begin{tabular}{lll}

299 \<open>\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x\<close> & and & \<open>\<forall>x \<in> H. h x \<le> p x\<close> \\

300 \end{tabular}

301 \end{center}

302 \<close>

304 theorem abs_Hahn_Banach:

305 assumes E: "vectorspace E" and FE: "subspace F E"

306 and lf: "linearform F f" and sn: "seminorm E p"

307 assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

308 shows "\<exists>g. linearform E g

309 \<and> (\<forall>x \<in> F. g x = f x)

310 \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"

311 proof -

312 interpret vectorspace E by fact

313 interpret subspace F E by fact

314 interpret linearform F f by fact

315 interpret seminorm E p by fact

316 have "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x) \<and> (\<forall>x \<in> E. g x \<le> p x)"

317 using E FE sn lf

318 proof (rule Hahn_Banach)

319 show "\<forall>x \<in> F. f x \<le> p x"

320 using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])

321 qed

322 then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"

323 and **: "\<forall>x \<in> E. g x \<le> p x" by blast

324 have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

325 using _ E sn lg **

326 proof (rule abs_ineq_iff [THEN iffD2])

327 show "E \<unlhd> E" ..

328 qed

329 with lg * show ?thesis by blast

330 qed

333 subsection \<open>The Hahn-Banach Theorem for normed spaces\<close>

335 text \<open>

336 Every continuous linear form \<open>f\<close> on a subspace \<open>F\<close> of a norm space \<open>E\<close>, can

337 be extended to a continuous linear form \<open>g\<close> on \<open>E\<close> such that \<open>\<parallel>f\<parallel> = \<parallel>g\<parallel>\<close>.

338 \<close>

340 theorem norm_Hahn_Banach:

341 fixes V and norm ("\<parallel>_\<parallel>")

342 fixes B defines "\<And>V f. B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"

343 fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)

344 defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"

345 assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"

346 and linearform: "linearform F f" and "continuous F f norm"

347 shows "\<exists>g. linearform E g

348 \<and> continuous E g norm

349 \<and> (\<forall>x \<in> F. g x = f x)

350 \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

351 proof -

352 interpret normed_vectorspace E norm by fact

353 interpret normed_vectorspace_with_fn_norm E norm B fn_norm

354 by (auto simp: B_def fn_norm_def) intro_locales

355 interpret subspace F E by fact

356 interpret linearform F f by fact

357 interpret continuous F f norm by fact

358 have E: "vectorspace E" by intro_locales

359 have F: "vectorspace F" by rule intro_locales

360 have F_norm: "normed_vectorspace F norm"

361 using FE E_norm by (rule subspace_normed_vs)

362 have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"

363 by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero

364 [OF normed_vectorspace_with_fn_norm.intro,

365 OF F_norm \<open>continuous F f norm\<close> , folded B_def fn_norm_def])

366 txt \<open>We define a function \<open>p\<close> on \<open>E\<close> as follows:

367 \<open>p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>\<close>

368 define p where "p x = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>" for x

370 txt \<open>\<open>p\<close> is a seminorm on \<open>E\<close>:\<close>

371 have q: "seminorm E p"

372 proof

373 fix x y a assume x: "x \<in> E" and y: "y \<in> E"

375 txt \<open>\<open>p\<close> is positive definite:\<close>

376 have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

377 moreover from x have "0 \<le> \<parallel>x\<parallel>" ..

378 ultimately show "0 \<le> p x"

379 by (simp add: p_def zero_le_mult_iff)

381 txt \<open>\<open>p\<close> is absolutely homogeneous:\<close>

383 show "p (a \<cdot> x) = \<bar>a\<bar> * p x"

384 proof -

385 have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)

386 also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)

387 also have "\<parallel>f\<parallel>\<hyphen>F * (\<bar>a\<bar> * \<parallel>x\<parallel>) = \<bar>a\<bar> * (\<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>)" by simp

388 also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)

389 finally show ?thesis .

390 qed

392 txt \<open>Furthermore, \<open>p\<close> is subadditive:\<close>

394 show "p (x + y) \<le> p x + p y"

395 proof -

396 have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)

397 also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)

398 from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..

399 with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"

400 by (simp add: mult_left_mono)

401 also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: distrib_left)

402 also have "\<dots> = p x + p y" by (simp only: p_def)

403 finally show ?thesis .

404 qed

405 qed

407 txt \<open>\<open>f\<close> is bounded by \<open>p\<close>.\<close>

409 have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"

410 proof

411 fix x assume "x \<in> F"

412 with \<open>continuous F f norm\<close> and linearform

413 show "\<bar>f x\<bar> \<le> p x"

414 unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong

415 [OF normed_vectorspace_with_fn_norm.intro,

416 OF F_norm, folded B_def fn_norm_def])

417 qed

419 txt \<open>Using the fact that \<open>p\<close> is a seminorm and \<open>f\<close> is bounded by \<open>p\<close> we can

420 apply the Hahn-Banach Theorem for real vector spaces. So \<open>f\<close> can be

421 extended in a norm-preserving way to some function \<open>g\<close> on the whole vector

422 space \<open>E\<close>.\<close>

424 with E FE linearform q obtain g where

425 linearformE: "linearform E g"

426 and a: "\<forall>x \<in> F. g x = f x"

427 and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"

428 by (rule abs_Hahn_Banach [elim_format]) iprover

430 txt \<open>We furthermore have to show that \<open>g\<close> is also continuous:\<close>

432 have g_cont: "continuous E g norm" using linearformE

433 proof

434 fix x assume "x \<in> E"

435 with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

436 by (simp only: p_def)

437 qed

439 txt \<open>To complete the proof, we show that \<open>\<parallel>g\<parallel> = \<parallel>f\<parallel>\<close>.\<close>

441 have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"

442 proof (rule order_antisym)

443 txt \<open>

444 First we show \<open>\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>\<close>. The function norm \<open>\<parallel>g\<parallel>\<close> is defined as the

445 smallest \<open>c \<in> \<real>\<close> such that

446 \begin{center}

447 \begin{tabular}{l}

448 \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>\<close>

449 \end{tabular}

450 \end{center}

451 \<^noindent> Furthermore holds

452 \begin{center}

453 \begin{tabular}{l}

454 \<open>\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>\<close>

455 \end{tabular}

456 \end{center}

457 \<close>

459 from g_cont _ ge_zero

460 show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"

461 proof

462 fix x assume "x \<in> E"

463 with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"

464 by (simp only: p_def)

465 qed

467 txt \<open>The other direction is achieved by a similar argument.\<close>

469 show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"

470 proof (rule normed_vectorspace_with_fn_norm.fn_norm_least

471 [OF normed_vectorspace_with_fn_norm.intro,

472 OF F_norm, folded B_def fn_norm_def])

473 fix x assume x: "x \<in> F"

474 show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

475 proof -

476 from a x have "g x = f x" ..

477 then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)

478 also from g_cont have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"

479 proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])

480 from FE x show "x \<in> E" ..

481 qed

482 finally show ?thesis .

483 qed

484 next

485 show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"

486 using g_cont by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])

487 show "continuous F f norm" by fact

488 qed

489 qed

490 with linearformE a g_cont show ?thesis by blast

491 qed

493 end