Say I have a complex polynomial:
$$a_0+a_1x+\cdots+a_nx^n,$$
where $a_0,\ldots,a_n$ are complex numbers.
What is the conjugate of this polynomial? How is it defined?
For example, if we have an inner product on the vector space $V$ defined by (for complex polynomials):
$$\int_0^1 p(x)\overline{q(x)}dx$$
Then how do we know that:
$$\langle v,v \rangle = \int_0^1 (a_0+a_1x+\cdots+a_nx^n)(\overline{a_0}+\overline{a_1}x+\cdots+\overline{a_n}x^n)dx$$
is positive? Since there could be negative numbers in there, do we just know the positive ones outweigh the negative?