Linear Algebra Done Right: why we define the inner product between two polynomials (of degree at most m) as $<p,q>=\int_{0}^1p(x)\bar{q}(x)dx$?

Question

In Linear Algebra Done Right (2nd Edition) equation (6.2), we defined the inner product between two polynomials as $<p,q>=\int_{0}^1p(x)\bar{q}(x)dx$. I'm wondering if there is a particular reason for defining the inner product between two polynomials (with degree at most $m$) as $<p,q>=\int_{0}^1p(x)\bar{q}(x)dx$ instead of $<p,q>=\int_{a}^bp(x)\bar{q}(x)dx$ for any constants $a,b$. Seems to me that $<p,q>=\int_{a}^bp(x)\bar{q}(x)dx$ is also a legitimate inner product.