1(* Title: HOL/Hahn_Banach/Hahn_Banach.thy 2 Author: Gertrud Bauer, TU Munich 3*) 4 5section \<open>The Hahn-Banach Theorem\<close> 6 7theory Hahn_Banach 8imports Hahn_Banach_Lemmas 9begin 10 11text \<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> 15 16 17subsection \<open>The Hahn-Banach Theorem for vector spaces\<close> 18 19paragraph \<open>Hahn-Banach Theorem.\<close> 20text \<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> 26 27paragraph \<open>Proof Sketch.\<close> 28text \<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. 31 32 \<^enum> We show that every non-empty chain in \<open>M\<close> has an upper bound in \<open>M\<close>. 33 34 \<^enum> With Zorn's Lemma we conclude that there is a maximal function \<open>g\<close> in \<open>M\<close>. 35 36 \<^enum> The domain \<open>H\<close> of \<open>g\<close> is the whole space \<open>E\<close>, as shown by classical 37 contradiction: 38 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>. 41 42 \<^item> Thus \<open>g\<close> can not be maximal. Contradiction! 43\<close> 44 45theorem 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> 53proof - 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)" 71 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) 122 123 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 142 143 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 150 151 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 180 181 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> 184 185 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 201 202 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 274 275 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 286qed 287 288 289subsection \<open>Alternative formulation\<close> 290 291text \<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> 303 304theorem 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)" 311proof - 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 330qed 331 332 333subsection \<open>The Hahn-Banach Theorem for normed spaces\<close> 334 335text \<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> 339 340theorem 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" 351proof - 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 369 370 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" 374 375 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) 380 381 txt \<open>\<open>p\<close> is absolutely homogeneous:\<close> 382 383 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 391 392 txt \<open>Furthermore, \<open>p\<close> is subadditive:\<close> 393 394 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 406 407 txt \<open>\<open>f\<close> is bounded by \<open>p\<close>.\<close> 408 409 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 418 419 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> 423 424 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 429 430 txt \<open>We furthermore have to show that \<open>g\<close> is also continuous:\<close> 431 432 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 438 439 txt \<open>To complete the proof, we show that \<open>\<parallel>g\<parallel> = \<parallel>f\<parallel>\<close>.\<close> 440 441 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> 458 459 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 466 467 txt \<open>The other direction is achieved by a similar argument.\<close> 468 469 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 491qed 492 493end 494