Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms. 1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a function? 2. What does "there is assigned a real number" suggest? Could someone please explain it to me with examples?