# HG changeset patch # User huffman # Date 1277918762 25200 # Node ID 5cbd5d5959f27fe401d2a08bce1a3e8a1414344a # Parent 48bdc2a43b46bb6333a0af82e7a40c33eee41255 generalize some euclidean_space lemmas diff -r 48bdc2a43b46 -r 5cbd5d5959f2 src/HOL/Multivariate_Analysis/Euclidean_Space.thy --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Jun 30 18:19:53 2010 +0200 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Jun 30 10:26:02 2010 -0700 @@ -1891,7 +1891,7 @@ subsection {* Linearity and Bilinearity continued *} lemma linear_bounded: - fixes f:: "'a::euclidean_space \ 'b::euclidean_space" + fixes f:: "'a::euclidean_space \ 'b::real_normed_vector" assumes lf: "linear f" shows "\B. \x. norm (f x) \ B * norm x" proof- @@ -1919,7 +1919,7 @@ qed lemma linear_bounded_pos: - fixes f:: "'a::euclidean_space \ 'b::euclidean_space" + fixes f:: "'a::euclidean_space \ 'b::real_normed_vector" assumes lf: "linear f" shows "\B > 0. \x. norm (f x) \ B * norm x" proof- @@ -1947,7 +1947,7 @@ qed lemma linear_conv_bounded_linear: - fixes f :: "'a::euclidean_space \ 'b::euclidean_space" + fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" shows "linear f \ bounded_linear f" proof assume "linear f" @@ -1971,14 +1971,14 @@ by (simp add: f.add f.scaleR linear_def) qed -lemma bounded_linearI': fixes f::"'a::euclidean_space \ 'b::euclidean_space" +lemma bounded_linearI': fixes f::"'a::euclidean_space \ 'b::real_normed_vector" assumes "\x y. f (x + y) = f x + f y" "\c x. f (c *\<^sub>R x) = c *\<^sub>R f x" shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym] by(rule linearI[OF assms]) lemma bilinear_bounded: - fixes h:: "'m::euclidean_space \ 'n::euclidean_space \ 'k::euclidean_space" + fixes h:: "'m::euclidean_space \ 'n::euclidean_space \ 'k::real_normed_vector" assumes bh: "bilinear h" shows "\B. \x y. norm (h x y) \ B * norm x * norm y" proof- @@ -2008,7 +2008,7 @@ qed lemma bilinear_bounded_pos: - fixes h:: "'a::euclidean_space \ 'b::euclidean_space \ 'c::euclidean_space" + fixes h:: "'a::euclidean_space \ 'b::euclidean_space \ 'c::real_normed_vector" assumes bh: "bilinear h" shows "\B > 0. \x y. norm (h x y) \ B * norm x * norm y" proof- @@ -2029,7 +2029,7 @@ qed lemma bilinear_conv_bounded_bilinear: - fixes h :: "'a::euclidean_space \ 'b::euclidean_space \ 'c::euclidean_space" + fixes h :: "'a::euclidean_space \ 'b::euclidean_space \ 'c::real_normed_vector" shows "bilinear h \ bounded_bilinear h" proof assume "bilinear h"