see the attachment, it is easy to show that is a inner product, but what condition needed to fulfill to show that forms a a Euclidian space?
The standard definition of "Euclidean space" is just for some n- that is, the set of ordered m-tuples of real numbers with both addition and scalar multiplication defined "coordinate-wise"- that is and .
There is a "standard basis" for . It consists of - that is, each basis vector has "1" in one place and "0" everywhere else.
There is also a "standard basis" for Pn, the space of polynomials of degree at most n: . The standard way to show that one vector space is isomorphic to another is to show that you can map one basis into the other. Be a little be careful here: while has dimension m, Pn has dimension n+1. For example, a basis for P2 is which contains three vectors and so P2 has dimension 3. To show that is isomorphic to , map 1 to (1, 0, 0), x to (0, 1, 0), and to (0, 0, 1). That would then map any to (a, b, c).