author huffman Sat, 21 Feb 2009 16:51:42 -0800 changeset 30046 49f603f92c47 parent 30045 b8ddd7667eed child 30048 6cf1fe60ac73
fix spelling
```--- a/src/HOL/Library/Inner_Product.thy	Sat Feb 21 15:39:59 2009 -0800
+++ b/src/HOL/Library/Inner_Product.thy	Sat Feb 21 16:51:42 2009 -0800
@@ -65,7 +65,7 @@
lemma power2_norm_eq_inner: "(norm x)\<twosuperior> = inner x x"

-lemma Cauchy_Schwartz_ineq:
+lemma Cauchy_Schwarz_ineq:
"(inner x y)\<twosuperior> \<le> inner x x * inner y y"
proof (cases)
assume "y = 0"
@@ -86,11 +86,11 @@
qed

-lemma Cauchy_Schwartz_ineq2:
+lemma Cauchy_Schwarz_ineq2:
"\<bar>inner x y\<bar> \<le> norm x * norm y"
proof (rule power2_le_imp_le)
have "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
-    using Cauchy_Schwartz_ineq .
+    using Cauchy_Schwarz_ineq .
thus "\<bar>inner x y\<bar>\<twosuperior> \<le> (norm x * norm y)\<twosuperior>"
show "0 \<le> norm x * norm y"
@@ -108,7 +108,7 @@
show "norm (x + y) \<le> norm x + norm y"
proof (rule power2_le_imp_le)
have "inner x y \<le> norm x * norm y"
-        by (rule order_trans [OF abs_ge_self Cauchy_Schwartz_ineq2])
+        by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2])
thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>"
unfolding power2_sum power2_norm_eq_inner