"Normalizing" a Function One of our homework problems for my linear algebra course this week is:
On $\mathcal{P}_2(\mathbb{R})$, consider the inner product given by
$$
\left<p, q\right> = \int_0^1 p(x)q(x) \, dx
$$
Apply the Gram-Schmidt procedure to the basis $\left\{1, x, x^2\right\}$ to produce an orthonormal basis for $\mathcal{P}_2(\mathbb{R})$.
Generating an orthogonal basis is trivial but I'm not quite sure how to go about normalizing the functions I get to "length one". For vectors, it's easy since you just divide by the magnitude but what about for functions?

The norm in this space is
$$\|u\| = \sqrt{\langle u, u\rangle} = \sqrt{\int_0^1 \left(u(x)\right)^2 dx}$$
So once you have a basis of three functions, compute the norms (i.e. compute the integral of the square, and square root it) and divide the function by the norm. In particular, show that
$$\left\| \frac{u}{\|u\|}\right\| = 1$$
