equal
deleted
inserted
replaced
8 imports Complex_Main FrechetDeriv |
8 imports Complex_Main FrechetDeriv |
9 begin |
9 begin |
10 |
10 |
11 subsection {* Real inner product spaces *} |
11 subsection {* Real inner product spaces *} |
12 |
12 |
13 class real_inner = real_vector + sgn_div_norm + |
13 class real_inner = real_vector + sgn_div_norm + dist_norm + |
14 fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real" |
14 fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real" |
15 assumes inner_commute: "inner x y = inner y x" |
15 assumes inner_commute: "inner x y = inner y x" |
16 and inner_left_distrib: "inner (x + y) z = inner x z + inner y z" |
16 and inner_left_distrib: "inner (x + y) z = inner x z + inner y z" |
17 and inner_scaleR_left: "inner (scaleR r x) y = r * (inner x y)" |
17 and inner_scaleR_left: "inner (scaleR r x) y = r * (inner x y)" |
18 and inner_ge_zero [simp]: "0 \<le> inner x x" |
18 and inner_ge_zero [simp]: "0 \<le> inner x x" |