Finding inner product and expression for norm

We define the functions, $f_n$, $n=0,1,2,3,...,$ on $[-1,1]$ by $$f_n(x)=x^n$$

Then I have to find the inner product for $\langle f_n,f_m \rangle$ for all values of $n, m$ and I have in particular, to show that: $$||f_n||=(n+\frac{1}{2})^{-\frac{1}{2}}$$

But I'm not sure how to do that? I'm confused, what is m? I think to show the norm I have to use $||f_n||=\sqrt{\langle f_n|f_n\rangle}$ ? But how do I find the inner product and use it finding the norm? I hope anyone can help me?

Solution 1:

Assuming you are using the standard $L^2$-inner product $\langle f, g\rangle = \int_{-1}^1 f(x)g(x)\, dx$, we have that \begin{align*} \|f_n\|^2 = \langle f_n, f_n\rangle = \int_{-1}^1 x^{2n}\, dx = \frac{x^{2n + 1}}{2n + 1}\bigg|_{-1}^1 = \frac{2}{2n + 1} = \frac{1}{(n + 1/2)}. \end{align*}