is inner product only valid for euclid space?

Question

is inner product only valid for euclidean space $\mathbb R^n$?

I mean, basic idea of inner product is that product of two elements in vector is real or complex.

So, if it is valid for other vector spaces such as polynomials, it does not seem like it works

Answer 1

Inner product is a bilinear funtion defined as here.

One interesting example as you mentioned is when $V=C^{0}([a,b])$, the vector space whose elements are continuous functions $g,f:[a,b] \rightarrow \mathbb{R}$. As an exercise you could show that

$$ \langle f,g \rangle = \int_{a}^{b} \ f(x)g(x)\ dx$$ is an inner product. In this case, the norm of the function $f$ is given by

$$|f| = \sqrt{\int_{a}^{b} \ f(x)^{2}\ dx}$$

This inner product is used in Fourier Series.

Answer 2

Actually, the inner product works fine for polynomials! Think of a polynomial $\sum a_i x^i$ as the vector of its coefficients, $(a_0...a_n)$. Then just take the ordinary inner product of two such vectors, adding enough zeroes on the end of the shorter one to make this make sense. You might think about whether this still works for the vector space of power series.