equal
deleted
inserted
replaced
151 by (simp add: inner_add inner_commute) |
151 by (simp add: inner_add inner_commute) |
152 show "0 \<le> norm x + norm y" |
152 show "0 \<le> norm x + norm y" |
153 unfolding norm_eq_sqrt_inner by simp |
153 unfolding norm_eq_sqrt_inner by simp |
154 qed |
154 qed |
155 have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)" |
155 have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)" |
156 by (simp add: real_sqrt_mult_distrib) |
156 by (simp add: real_sqrt_mult) |
157 then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x" |
157 then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x" |
158 unfolding norm_eq_sqrt_inner |
158 unfolding norm_eq_sqrt_inner |
159 by (simp add: power2_eq_square mult.assoc) |
159 by (simp add: power2_eq_square mult.assoc) |
160 qed |
160 qed |
161 |
161 |