1.1 --- a/src/HOL/HahnBanach/HahnBanach.thy Wed Jun 24 21:28:02 2009 +0200
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,509 +0,0 @@
1.4 -(* Title: HOL/Real/HahnBanach/HahnBanach.thy
1.5 - Author: Gertrud Bauer, TU Munich
1.6 -*)
1.7 -
1.8 -header {* The Hahn-Banach Theorem *}
1.9 -
1.10 -theory HahnBanach
1.11 -imports HahnBanachLemmas
1.12 -begin
1.13 -
1.14 -text {*
1.15 - We present the proof of two different versions of the Hahn-Banach
1.16 - Theorem, closely following \cite[\S36]{Heuser:1986}.
1.17 -*}
1.18 -
1.19 -subsection {* The Hahn-Banach Theorem for vector spaces *}
1.20 -
1.21 -text {*
1.22 - \textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real
1.23 - vector space @{text E}, let @{text p} be a semi-norm on @{text E},
1.24 - and @{text f} be a linear form defined on @{text F} such that @{text
1.25 - f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in> F. f x \<le> p x"}. Then
1.26 - @{text f} can be extended to a linear form @{text h} on @{text E}
1.27 - such that @{text h} is norm-preserving, i.e. @{text h} is also
1.28 - bounded by @{text p}.
1.29 -
1.30 - \bigskip
1.31 - \textbf{Proof Sketch.}
1.32 - \begin{enumerate}
1.33 -
1.34 - \item Define @{text M} as the set of norm-preserving extensions of
1.35 - @{text f} to subspaces of @{text E}. The linear forms in @{text M}
1.36 - are ordered by domain extension.
1.37 -
1.38 - \item We show that every non-empty chain in @{text M} has an upper
1.39 - bound in @{text M}.
1.40 -
1.41 - \item With Zorn's Lemma we conclude that there is a maximal function
1.42 - @{text g} in @{text M}.
1.43 -
1.44 - \item The domain @{text H} of @{text g} is the whole space @{text
1.45 - E}, as shown by classical contradiction:
1.46 -
1.47 - \begin{itemize}
1.48 -
1.49 - \item Assuming @{text g} is not defined on whole @{text E}, it can
1.50 - still be extended in a norm-preserving way to a super-space @{text
1.51 - H'} of @{text H}.
1.52 -
1.53 - \item Thus @{text g} can not be maximal. Contradiction!
1.54 -
1.55 - \end{itemize}
1.56 - \end{enumerate}
1.57 -*}
1.58 -
1.59 -theorem HahnBanach:
1.60 - assumes E: "vectorspace E" and "subspace F E"
1.61 - and "seminorm E p" and "linearform F f"
1.62 - assumes fp: "\<forall>x \<in> F. f x \<le> p x"
1.63 - 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)"
1.64 - -- {* Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E}, *}
1.65 - -- {* and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p}, *}
1.66 - -- {* then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \skp *}
1.67 -proof -
1.68 - interpret vectorspace E by fact
1.69 - interpret subspace F E by fact
1.70 - interpret seminorm E p by fact
1.71 - interpret linearform F f by fact
1.72 - def M \<equiv> "norm_pres_extensions E p F f"
1.73 - then have M: "M = \<dots>" by (simp only:)
1.74 - from E have F: "vectorspace F" ..
1.75 - note FE = `F \<unlhd> E`
1.76 - {
1.77 - fix c assume cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c"
1.78 - have "\<Union>c \<in> M"
1.79 - -- {* Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}: *}
1.80 - -- {* @{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}. *}
1.81 - unfolding M_def
1.82 - proof (rule norm_pres_extensionI)
1.83 - let ?H = "domain (\<Union>c)"
1.84 - let ?h = "funct (\<Union>c)"
1.85 -
1.86 - have a: "graph ?H ?h = \<Union>c"
1.87 - proof (rule graph_domain_funct)
1.88 - fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"
1.89 - with M_def cM show "z = y" by (rule sup_definite)
1.90 - qed
1.91 - moreover from M cM a have "linearform ?H ?h"
1.92 - by (rule sup_lf)
1.93 - moreover from a M cM ex FE E have "?H \<unlhd> E"
1.94 - by (rule sup_subE)
1.95 - moreover from a M cM ex FE have "F \<unlhd> ?H"
1.96 - by (rule sup_supF)
1.97 - moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
1.98 - by (rule sup_ext)
1.99 - moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"
1.100 - by (rule sup_norm_pres)
1.101 - ultimately show "\<exists>H h. \<Union>c = graph H h
1.102 - \<and> linearform H h
1.103 - \<and> H \<unlhd> E
1.104 - \<and> F \<unlhd> H
1.105 - \<and> graph F f \<subseteq> graph H h
1.106 - \<and> (\<forall>x \<in> H. h x \<le> p x)" by blast
1.107 - qed
1.108 - }
1.109 - then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
1.110 - -- {* With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp *}
1.111 - proof (rule Zorn's_Lemma)
1.112 - -- {* We show that @{text M} is non-empty: *}
1.113 - show "graph F f \<in> M"
1.114 - unfolding M_def
1.115 - proof (rule norm_pres_extensionI2)
1.116 - show "linearform F f" by fact
1.117 - show "F \<unlhd> E" by fact
1.118 - from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)
1.119 - show "graph F f \<subseteq> graph F f" ..
1.120 - show "\<forall>x\<in>F. f x \<le> p x" by fact
1.121 - qed
1.122 - qed
1.123 - then obtain g where gM: "g \<in> M" and gx: "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
1.124 - by blast
1.125 - from gM obtain H h where
1.126 - g_rep: "g = graph H h"
1.127 - and linearform: "linearform H h"
1.128 - and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"
1.129 - and graphs: "graph F f \<subseteq> graph H h"
1.130 - and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..
1.131 - -- {* @{text g} is a norm-preserving extension of @{text f}, in other words: *}
1.132 - -- {* @{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E}, *}
1.133 - -- {* and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp *}
1.134 - from HE E have H: "vectorspace H"
1.135 - by (rule subspace.vectorspace)
1.136 -
1.137 - have HE_eq: "H = E"
1.138 - -- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}
1.139 - proof (rule classical)
1.140 - assume neq: "H \<noteq> E"
1.141 - -- {* Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended *}
1.142 - -- {* in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp *}
1.143 - have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
1.144 - proof -
1.145 - from HE have "H \<subseteq> E" ..
1.146 - with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast
1.147 - obtain x': "x' \<noteq> 0"
1.148 - proof
1.149 - show "x' \<noteq> 0"
1.150 - proof
1.151 - assume "x' = 0"
1.152 - with H have "x' \<in> H" by (simp only: vectorspace.zero)
1.153 - with `x' \<notin> H` show False by contradiction
1.154 - qed
1.155 - qed
1.156 -
1.157 - def H' \<equiv> "H + lin x'"
1.158 - -- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}
1.159 - have HH': "H \<unlhd> H'"
1.160 - proof (unfold H'_def)
1.161 - from x'E have "vectorspace (lin x')" ..
1.162 - with H show "H \<unlhd> H + lin x'" ..
1.163 - qed
1.164 -
1.165 - obtain xi where
1.166 - xi: "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
1.167 - \<and> xi \<le> p (y + x') - h y"
1.168 - -- {* Pick a real number @{text \<xi>} that fulfills certain inequations; this will *}
1.169 - -- {* be used to establish that @{text h'} is a norm-preserving extension of @{text h}.
1.170 - \label{ex-xi-use}\skp *}
1.171 - proof -
1.172 - from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
1.173 - \<and> xi \<le> p (y + x') - h y"
1.174 - proof (rule ex_xi)
1.175 - fix u v assume u: "u \<in> H" and v: "v \<in> H"
1.176 - with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto
1.177 - from H u v linearform have "h v - h u = h (v - u)"
1.178 - by (simp add: linearform.diff)
1.179 - also from hp and H u v have "\<dots> \<le> p (v - u)"
1.180 - by (simp only: vectorspace.diff_closed)
1.181 - also from x'E uE vE have "v - u = x' + - x' + v + - u"
1.182 - by (simp add: diff_eq1)
1.183 - also from x'E uE vE have "\<dots> = v + x' + - (u + x')"
1.184 - by (simp add: add_ac)
1.185 - also from x'E uE vE have "\<dots> = (v + x') - (u + x')"
1.186 - by (simp add: diff_eq1)
1.187 - also from x'E uE vE E have "p \<dots> \<le> p (v + x') + p (u + x')"
1.188 - by (simp add: diff_subadditive)
1.189 - finally have "h v - h u \<le> p (v + x') + p (u + x')" .
1.190 - then show "- p (u + x') - h u \<le> p (v + x') - h v" by simp
1.191 - qed
1.192 - then show thesis by (blast intro: that)
1.193 - qed
1.194 -
1.195 - def h' \<equiv> "\<lambda>x. let (y, a) =
1.196 - SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"
1.197 - -- {* Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp *}
1.198 -
1.199 - have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
1.200 - -- {* @{text h'} is an extension of @{text h} \dots \skp *}
1.201 - proof
1.202 - show "g \<subseteq> graph H' h'"
1.203 - proof -
1.204 - have "graph H h \<subseteq> graph H' h'"
1.205 - proof (rule graph_extI)
1.206 - fix t assume t: "t \<in> H"
1.207 - from E HE t have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
1.208 - using `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` by (rule decomp_H'_H)
1.209 - with h'_def show "h t = h' t" by (simp add: Let_def)
1.210 - next
1.211 - from HH' show "H \<subseteq> H'" ..
1.212 - qed
1.213 - with g_rep show ?thesis by (simp only:)
1.214 - qed
1.215 -
1.216 - show "g \<noteq> graph H' h'"
1.217 - proof -
1.218 - have "graph H h \<noteq> graph H' h'"
1.219 - proof
1.220 - assume eq: "graph H h = graph H' h'"
1.221 - have "x' \<in> H'"
1.222 - unfolding H'_def
1.223 - proof
1.224 - from H show "0 \<in> H" by (rule vectorspace.zero)
1.225 - from x'E show "x' \<in> lin x'" by (rule x_lin_x)
1.226 - from x'E show "x' = 0 + x'" by simp
1.227 - qed
1.228 - then have "(x', h' x') \<in> graph H' h'" ..
1.229 - with eq have "(x', h' x') \<in> graph H h" by (simp only:)
1.230 - then have "x' \<in> H" ..
1.231 - with `x' \<notin> H` show False by contradiction
1.232 - qed
1.233 - with g_rep show ?thesis by simp
1.234 - qed
1.235 - qed
1.236 - moreover have "graph H' h' \<in> M"
1.237 - -- {* and @{text h'} is norm-preserving. \skp *}
1.238 - proof (unfold M_def)
1.239 - show "graph H' h' \<in> norm_pres_extensions E p F f"
1.240 - proof (rule norm_pres_extensionI2)
1.241 - show "linearform H' h'"
1.242 - using h'_def H'_def HE linearform `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E
1.243 - by (rule h'_lf)
1.244 - show "H' \<unlhd> E"
1.245 - unfolding H'_def
1.246 - proof
1.247 - show "H \<unlhd> E" by fact
1.248 - show "vectorspace E" by fact
1.249 - from x'E show "lin x' \<unlhd> E" ..
1.250 - qed
1.251 - from H `F \<unlhd> H` HH' show FH': "F \<unlhd> H'"
1.252 - by (rule vectorspace.subspace_trans)
1.253 - show "graph F f \<subseteq> graph H' h'"
1.254 - proof (rule graph_extI)
1.255 - fix x assume x: "x \<in> F"
1.256 - with graphs have "f x = h x" ..
1.257 - also have "\<dots> = h x + 0 * xi" by simp
1.258 - also have "\<dots> = (let (y, a) = (x, 0) in h y + a * xi)"
1.259 - by (simp add: Let_def)
1.260 - also have "(x, 0) =
1.261 - (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)"
1.262 - using E HE
1.263 - proof (rule decomp_H'_H [symmetric])
1.264 - from FH x show "x \<in> H" ..
1.265 - from x' show "x' \<noteq> 0" .
1.266 - show "x' \<notin> H" by fact
1.267 - show "x' \<in> E" by fact
1.268 - qed
1.269 - also have
1.270 - "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)
1.271 - in h y + a * xi) = h' x" by (simp only: h'_def)
1.272 - finally show "f x = h' x" .
1.273 - next
1.274 - from FH' show "F \<subseteq> H'" ..
1.275 - qed
1.276 - show "\<forall>x \<in> H'. h' x \<le> p x"
1.277 - using h'_def H'_def `x' \<notin> H` `x' \<in> E` `x' \<noteq> 0` E HE
1.278 - `seminorm E p` linearform and hp xi
1.279 - by (rule h'_norm_pres)
1.280 - qed
1.281 - qed
1.282 - ultimately show ?thesis ..
1.283 - qed
1.284 - then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
1.285 - -- {* So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp *}
1.286 - with gx show "H = E" by contradiction
1.287 - qed
1.288 -
1.289 - from HE_eq and linearform have "linearform E h"
1.290 - by (simp only:)
1.291 - moreover have "\<forall>x \<in> F. h x = f x"
1.292 - proof
1.293 - fix x assume "x \<in> F"
1.294 - with graphs have "f x = h x" ..
1.295 - then show "h x = f x" ..
1.296 - qed
1.297 - moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
1.298 - by (simp only:)
1.299 - ultimately show ?thesis by blast
1.300 -qed
1.301 -
1.302 -
1.303 -subsection {* Alternative formulation *}
1.304 -
1.305 -text {*
1.306 - The following alternative formulation of the Hahn-Banach
1.307 - Theorem\label{abs-HahnBanach} uses the fact that for a real linear
1.308 - form @{text f} and a seminorm @{text p} the following inequations
1.309 - are equivalent:\footnote{This was shown in lemma @{thm [source]
1.310 - abs_ineq_iff} (see page \pageref{abs-ineq-iff}).}
1.311 - \begin{center}
1.312 - \begin{tabular}{lll}
1.313 - @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
1.314 - @{text "\<forall>x \<in> H. h x \<le> p x"} \\
1.315 - \end{tabular}
1.316 - \end{center}
1.317 -*}
1.318 -
1.319 -theorem abs_HahnBanach:
1.320 - assumes E: "vectorspace E" and FE: "subspace F E"
1.321 - and lf: "linearform F f" and sn: "seminorm E p"
1.322 - assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
1.323 - shows "\<exists>g. linearform E g
1.324 - \<and> (\<forall>x \<in> F. g x = f x)
1.325 - \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
1.326 -proof -
1.327 - interpret vectorspace E by fact
1.328 - interpret subspace F E by fact
1.329 - interpret linearform F f by fact
1.330 - interpret seminorm E p by fact
1.331 - 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)"
1.332 - using E FE sn lf
1.333 - proof (rule HahnBanach)
1.334 - show "\<forall>x \<in> F. f x \<le> p x"
1.335 - using FE E sn lf and fp by (rule abs_ineq_iff [THEN iffD1])
1.336 - qed
1.337 - then obtain g where lg: "linearform E g" and *: "\<forall>x \<in> F. g x = f x"
1.338 - and **: "\<forall>x \<in> E. g x \<le> p x" by blast
1.339 - have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
1.340 - using _ E sn lg **
1.341 - proof (rule abs_ineq_iff [THEN iffD2])
1.342 - show "E \<unlhd> E" ..
1.343 - qed
1.344 - with lg * show ?thesis by blast
1.345 -qed
1.346 -
1.347 -
1.348 -subsection {* The Hahn-Banach Theorem for normed spaces *}
1.349 -
1.350 -text {*
1.351 - Every continuous linear form @{text f} on a subspace @{text F} of a
1.352 - norm space @{text E}, can be extended to a continuous linear form
1.353 - @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}.
1.354 -*}
1.355 -
1.356 -theorem norm_HahnBanach:
1.357 - fixes V and norm ("\<parallel>_\<parallel>")
1.358 - 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}"
1.359 - fixes fn_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
1.360 - defines "\<And>V f. \<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
1.361 - assumes E_norm: "normed_vectorspace E norm" and FE: "subspace F E"
1.362 - and linearform: "linearform F f" and "continuous F norm f"
1.363 - shows "\<exists>g. linearform E g
1.364 - \<and> continuous E norm g
1.365 - \<and> (\<forall>x \<in> F. g x = f x)
1.366 - \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
1.367 -proof -
1.368 - interpret normed_vectorspace E norm by fact
1.369 - interpret normed_vectorspace_with_fn_norm E norm B fn_norm
1.370 - by (auto simp: B_def fn_norm_def) intro_locales
1.371 - interpret subspace F E by fact
1.372 - interpret linearform F f by fact
1.373 - interpret continuous F norm f by fact
1.374 - have E: "vectorspace E" by intro_locales
1.375 - have F: "vectorspace F" by rule intro_locales
1.376 - have F_norm: "normed_vectorspace F norm"
1.377 - using FE E_norm by (rule subspace_normed_vs)
1.378 - have ge_zero: "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
1.379 - by (rule normed_vectorspace_with_fn_norm.fn_norm_ge_zero
1.380 - [OF normed_vectorspace_with_fn_norm.intro,
1.381 - OF F_norm `continuous F norm f` , folded B_def fn_norm_def])
1.382 - txt {* We define a function @{text p} on @{text E} as follows:
1.383 - @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"} *}
1.384 - def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
1.385 -
1.386 - txt {* @{text p} is a seminorm on @{text E}: *}
1.387 - have q: "seminorm E p"
1.388 - proof
1.389 - fix x y a assume x: "x \<in> E" and y: "y \<in> E"
1.390 -
1.391 - txt {* @{text p} is positive definite: *}
1.392 - have "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
1.393 - moreover from x have "0 \<le> \<parallel>x\<parallel>" ..
1.394 - ultimately show "0 \<le> p x"
1.395 - by (simp add: p_def zero_le_mult_iff)
1.396 -
1.397 - txt {* @{text p} is absolutely homogenous: *}
1.398 -
1.399 - show "p (a \<cdot> x) = \<bar>a\<bar> * p x"
1.400 - proof -
1.401 - have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>" by (simp only: p_def)
1.402 - also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>" by (rule abs_homogenous)
1.403 - 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
1.404 - also have "\<dots> = \<bar>a\<bar> * p x" by (simp only: p_def)
1.405 - finally show ?thesis .
1.406 - qed
1.407 -
1.408 - txt {* Furthermore, @{text p} is subadditive: *}
1.409 -
1.410 - show "p (x + y) \<le> p x + p y"
1.411 - proof -
1.412 - have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>" by (simp only: p_def)
1.413 - also have a: "0 \<le> \<parallel>f\<parallel>\<hyphen>F" by (rule ge_zero)
1.414 - from x y have "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
1.415 - with a have " \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
1.416 - by (simp add: mult_left_mono)
1.417 - also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>" by (simp only: right_distrib)
1.418 - also have "\<dots> = p x + p y" by (simp only: p_def)
1.419 - finally show ?thesis .
1.420 - qed
1.421 - qed
1.422 -
1.423 - txt {* @{text f} is bounded by @{text p}. *}
1.424 -
1.425 - have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
1.426 - proof
1.427 - fix x assume "x \<in> F"
1.428 - with `continuous F norm f` and linearform
1.429 - show "\<bar>f x\<bar> \<le> p x"
1.430 - unfolding p_def by (rule normed_vectorspace_with_fn_norm.fn_norm_le_cong
1.431 - [OF normed_vectorspace_with_fn_norm.intro,
1.432 - OF F_norm, folded B_def fn_norm_def])
1.433 - qed
1.434 -
1.435 - txt {* Using the fact that @{text p} is a seminorm and @{text f} is bounded
1.436 - by @{text p} we can apply the Hahn-Banach Theorem for real vector
1.437 - spaces. So @{text f} can be extended in a norm-preserving way to
1.438 - some function @{text g} on the whole vector space @{text E}. *}
1.439 -
1.440 - with E FE linearform q obtain g where
1.441 - linearformE: "linearform E g"
1.442 - and a: "\<forall>x \<in> F. g x = f x"
1.443 - and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
1.444 - by (rule abs_HahnBanach [elim_format]) iprover
1.445 -
1.446 - txt {* We furthermore have to show that @{text g} is also continuous: *}
1.447 -
1.448 - have g_cont: "continuous E norm g" using linearformE
1.449 - proof
1.450 - fix x assume "x \<in> E"
1.451 - with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
1.452 - by (simp only: p_def)
1.453 - qed
1.454 -
1.455 - txt {* To complete the proof, we show that @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. *}
1.456 -
1.457 - have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
1.458 - proof (rule order_antisym)
1.459 - txt {*
1.460 - First we show @{text "\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>"}. The function norm @{text
1.461 - "\<parallel>g\<parallel>"} is defined as the smallest @{text "c \<in> \<real>"} such that
1.462 - \begin{center}
1.463 - \begin{tabular}{l}
1.464 - @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
1.465 - \end{tabular}
1.466 - \end{center}
1.467 - \noindent Furthermore holds
1.468 - \begin{center}
1.469 - \begin{tabular}{l}
1.470 - @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
1.471 - \end{tabular}
1.472 - \end{center}
1.473 - *}
1.474 -
1.475 - have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
1.476 - proof
1.477 - fix x assume "x \<in> E"
1.478 - with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
1.479 - by (simp only: p_def)
1.480 - qed
1.481 - from g_cont this ge_zero
1.482 - show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
1.483 - by (rule fn_norm_least [of g, folded B_def fn_norm_def])
1.484 -
1.485 - txt {* The other direction is achieved by a similar argument. *}
1.486 -
1.487 - show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
1.488 - proof (rule normed_vectorspace_with_fn_norm.fn_norm_least
1.489 - [OF normed_vectorspace_with_fn_norm.intro,
1.490 - OF F_norm, folded B_def fn_norm_def])
1.491 - show "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
1.492 - proof
1.493 - fix x assume x: "x \<in> F"
1.494 - from a x have "g x = f x" ..
1.495 - then have "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
1.496 - also from g_cont
1.497 - have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
1.498 - proof (rule fn_norm_le_cong [OF _ linearformE, folded B_def fn_norm_def])
1.499 - from FE x show "x \<in> E" ..
1.500 - qed
1.501 - finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
1.502 - qed
1.503 - show "0 \<le> \<parallel>g\<parallel>\<hyphen>E"
1.504 - using g_cont
1.505 - by (rule fn_norm_ge_zero [of g, folded B_def fn_norm_def])
1.506 - show "continuous F norm f" by fact
1.507 - qed
1.508 - qed
1.509 - with linearformE a g_cont show ?thesis by blast
1.510 -qed
1.511 -
1.512 -end