I am given the space $P_3$ with inner product $\int_{0}^{2}f(x)g(x)dx$. I am asked to compute the length of $x$. Since the length of a vector is $\sqrt{x*x}$, I have set this up as $\sqrt{\int_{0}^{2}x^2dx}$. This does not seem right to me, though. Since the inner portion of the integral is $f(x)g(x)$, do I multiply $x$ by $x$ and then integrate? Or am I completely wrong in my approach to this?
Thank you!