--- a/src/HOL/Hahn_Banach/Hahn_Banach.thy Mon Oct 19 17:19:53 2015 +0200
+++ b/src/HOL/Hahn_Banach/Hahn_Banach.thy Mon Oct 19 17:45:36 2015 +0200
@@ -15,42 +15,36 @@
subsection \<open>The Hahn-Banach Theorem for vector spaces\<close>
+paragraph \<open>Hahn-Banach Theorem.\<close>
text \<open>
- \textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real
- vector space @{text E}, let @{text p} be a semi-norm on @{text E},
- and @{text f} be a linear form defined on @{text F} such that @{text
- f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in> F. f x \<le> p x"}. Then
- @{text f} can be extended to a linear form @{text h} on @{text E}
- such that @{text h} is norm-preserving, i.e. @{text h} is also
- bounded by @{text p}.
+ Let @{text F} be a subspace of a real vector space @{text E}, let @{text
+ p} be a semi-norm on @{text E}, and @{text f} be a linear form defined on
+ @{text F} such that @{text f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in>
+ F. f x \<le> p x"}. Then @{text f} can be extended to a linear form @{text h}
+ on @{text E} such that @{text h} is norm-preserving, i.e. @{text h} is
+ also bounded by @{text p}.
+\<close>
- \bigskip
- \textbf{Proof Sketch.}
- \begin{enumerate}
-
- \item Define @{text M} as the set of norm-preserving extensions of
+paragraph \<open>Proof Sketch.\<close>
+text \<open>
+ \<^enum> Define @{text M} as the set of norm-preserving extensions of
@{text f} to subspaces of @{text E}. The linear forms in @{text M}
are ordered by domain extension.
- \item We show that every non-empty chain in @{text M} has an upper
+ \<^enum> We show that every non-empty chain in @{text M} has an upper
bound in @{text M}.
- \item With Zorn's Lemma we conclude that there is a maximal function
+ \<^enum> With Zorn's Lemma we conclude that there is a maximal function
@{text g} in @{text M}.
- \item The domain @{text H} of @{text g} is the whole space @{text
+ \<^enum> The domain @{text H} of @{text g} is the whole space @{text
E}, as shown by classical contradiction:
- \begin{itemize}
+ \<^item> Assuming @{text g} is not defined on whole @{text E}, it can
+ still be extended in a norm-preserving way to a super-space @{text
+ H'} of @{text H}.
- \item Assuming @{text g} is not defined on whole @{text E}, it can
- still be extended in a norm-preserving way to a super-space @{text
- H'} of @{text H}.
-
- \item Thus @{text g} can not be maximal. Contradiction!
-
- \end{itemize}
- \end{enumerate}
+ \<^item> Thus @{text g} can not be maximal. Contradiction!
\<close>
theorem Hahn_Banach:
@@ -60,7 +54,7 @@
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)"
-- \<open>Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E},\<close>
-- \<open>and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p},\<close>
- -- \<open>then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \skp\<close>
+ -- \<open>then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \<^smallskip>\<close>
proof -
interpret vectorspace E by fact
interpret subspace F E by fact
@@ -104,7 +98,7 @@
qed
}
then have "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> x = g"
- -- \<open>With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp\<close>
+ -- \<open>With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \<^smallskip>\<close>
proof (rule Zorn's_Lemma)
-- \<open>We show that @{text M} is non-empty:\<close>
show "graph F f \<in> M"
@@ -127,16 +121,16 @@
and hp: "\<forall>x \<in> H. h x \<le> p x" unfolding M_def ..
-- \<open>@{text g} is a norm-preserving extension of @{text f}, in other words:\<close>
-- \<open>@{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E},\<close>
- -- \<open>and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp\<close>
+ -- \<open>and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \<^smallskip>\<close>
from HE E have H: "vectorspace H"
by (rule subspace.vectorspace)
have HE_eq: "H = E"
- -- \<open>We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp\<close>
+ -- \<open>We show that @{text h} is defined on whole @{text E} by classical contradiction. \<^smallskip>\<close>
proof (rule classical)
assume neq: "H \<noteq> E"
-- \<open>Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended\<close>
- -- \<open>in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp\<close>
+ -- \<open>in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \<^smallskip>\<close>
have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
proof -
from HE have "H \<subseteq> E" ..
@@ -152,7 +146,7 @@
qed
def H' \<equiv> "H + lin x'"
- -- \<open>Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp\<close>
+ -- \<open>Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \<^smallskip>\<close>
have HH': "H \<unlhd> H'"
proof (unfold H'_def)
from x'E have "vectorspace (lin x')" ..
@@ -164,7 +158,7 @@
\<and> xi \<le> p (y + x') - h y"
-- \<open>Pick a real number @{text \<xi>} that fulfills certain inequations; this will\<close>
-- \<open>be used to establish that @{text h'} is a norm-preserving extension of @{text h}.
- \label{ex-xi-use}\skp\<close>
+ \label{ex-xi-use}\<^smallskip>\<close>
proof -
from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
\<and> xi \<le> p (y + x') - h y"
@@ -191,10 +185,10 @@
def h' \<equiv> "\<lambda>x. let (y, a) =
SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"
- -- \<open>Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp\<close>
+ -- \<open>Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \<^smallskip>\<close>
have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
- -- \<open>@{text h'} is an extension of @{text h} \dots \skp\<close>
+ -- \<open>@{text h'} is an extension of @{text h} \dots \<^smallskip>\<close>
proof
show "g \<subseteq> graph H' h'"
proof -
@@ -231,7 +225,7 @@
qed
qed
moreover have "graph H' h' \<in> M"
- -- \<open>and @{text h'} is norm-preserving. \skp\<close>
+ -- \<open>and @{text h'} is norm-preserving. \<^smallskip>\<close>
proof (unfold M_def)
show "graph H' h' \<in> norm_pres_extensions E p F f"
proof (rule norm_pres_extensionI2)
@@ -279,7 +273,7 @@
ultimately show ?thesis ..
qed
then have "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
- -- \<open>So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp\<close>
+ -- \<open>So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \<^smallskip>\<close>
with gx show "H = E" by contradiction
qed