I'm having trouble with a couple of things written in some notes I'm reading. Firstly, in stating examples of vector spaces, they say Trigonometric polynomials - Given n distinct (mod 2π) complex constants λ1,...,λn, the set of all linear combinations of eiλnz forms an n-dimensional complex vector space. Now does this even make sense? I feel as though it is trying to say that we can have eiλnz for integer λn as the basis for the vector space of functions, i.e like Fourier series, but I'm not sure really. Secondly, for the complex vector space defined by the vectors in the 2D plane |r,φ>=(rcosφ,rsinφ) where vector addition is as usual and scalar multiplication follows the rule α|r,φ>=||α|r,φ+argα> I'm having trouble proving that α(|a>+|b>)=α|a>+α|b> and (α+β)|a>=α|a>+β|a> For the second one, I do (α+β)|a>=(α+β)|r,φ> =||(α+β)|r,φ+arg(α+β)> and I have no idea what to do next, as expanding the modulus/argument doesn't seem possible. Not sure about the second either.