Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm.
I tried to look at a general polynomial $\sum_{i=0}^{98} a_ix^i$ and use $\|f-g\|=\int_{-1}^1 f(x)\overline{g(x)} \, dx$ but this is too excessive and I can't see what it hinders. I also tried using the zero element but would end up with a non-zero result which isn't helpful much (if the questioned had asked about $x^{99}$ instead, I would have none that $\|x^{99}\|=0$ and there is no non-negative value smaller than 0). But this is not the case, so what should I do?
Edit: how about $x\in P_{\le 98}$? $\displaystyle\int_{-1}^1 x^{100} \, dx = \left.\frac{x^{102}} {102}\right|_{-1}^1=0$. Is it correct?
This is an elaboration on angryavian's comment.
The $L^2$-distance is defined by $$\|f-g\|_2=\left(\int_{-1}^1|f(z)-g(z)|^2\,dz\right)^{1/2}$$ with inner product given by $$\langle f,g\rangle_2=\int_{-1}^1f(x)\overline{g(x)}\,dx.$$
You will want to work with projection onto $P_{\le 98}$. Since we are working with finite-dimensional spaces, the point in $P_{\le 98}$ that is closest to some other point is the projection onto this subspace. Clearly, $P_{\le 98}$ is spanned by elements of the form $x^n$ for $n=0,1,\dots, 98$. So, you want to find a monic, degree 100 polynomial $P$ with zero coefficient of $x^{99}$ and such that $\int_{-1}^1P(x)x^n\,dx=0$ for each $n=0,\dots, 98$. This is just a system of 99 linear equations in 99 variables.
The point of $P_{98}$ which is closest to $x^{100}$ is $Q(x)=P(x)-x^{100}$, so you just find the distance between $Q$ and $x^{100}$.
In your edit, you integrated wrong and forgot to square the integrand.