hello, after some struggle,i finally understand these notations, but there is a problem. i'm studying with "principles of quantum mechanics" [R. Shankar]. in the beginning of the chapter where he discusses 'bra-kets' he mentioned the axioms of inner product. one of the axioms he mentioned [and used] is the following: -<V|aW>=a<V|W> [where 'a' is a complex scalar, V and W are vectors] My problem is this axiom assumes linearity with respect to the second term in the inner product, when until this book i encountered only with axiom who assumes linearity with respect to the first factor. after reading why it is so comfortable to define the 'bra' space, i can understand the logic of the author, but it seems strange to me that he changed the axiom a little bit for the purposes of quantum mechanics. Can anyone explain me what i have missed?