1. The problem statement, all variables and given/known data (i) Set of all row vectors: (a1,....,an), aj in K; addition, multiplication defined componentwise. This space is denoted as Kn. (ii) Set of all real valued functions f(x) defined on the real line, K = R. (iii) Set of all functions with values in K, defined on an arbitrary set S. (iv) Set of all polynomials of degree less than n with coefficients in K. 2. Relevant equations 1) Show that (i) and (iv) are isomorphic 2) Show that if S has n elements, (i) is the same as (iii) 3) Show that when K = R, (iv) is isomorphic with (iii) when S consists of n distinct points of R. 3. The attempt at a solution I've solved 1), but I cannot solve others. I think that problem is that I don't understand definition of (iii). Could someone please help me?