Approximate $f(x)= x^2$ with the best approximation in $L_1$ under the given norm.

Question

The given norm is: $||f||= \sqrt{A_0 |f(x_0)|^2 + A_1 |f(x_1)|^2 + A_2 |f(x_2)|^2}$ And from a previous question I have found the following:

$A_0 = \frac {-\sqrt{\pi/3}}{2}$

$A_1 = \frac {\sqrt{\pi/3}}{2}$

$A_2 = \sqrt{\pi}$

$x_0 = \sqrt{\frac{3}{2}}$

$x_1 = -\sqrt{\frac{3}{2}}$

$x_2 = -\sqrt{\frac{1}{2}}$

Now my plan of work is the following:

1. Choosing an orthogonal basis
2. Identifying the induced inner product
3. Building the system of equations and solving it

My problem is finding the inner product induced from a norm, I know how to find the norm of an inner product but not the opposite. Maybe there’s a different way to solve the question that differs from my plan of work? Any help would be appreciated.

Answer 1

First of all, a norm doesn't always arise from an inner product. It is the case though if the norm satisfies the parallelogram law. In that case, the inner product can be derived from the norm using the polarization identity. That is for real normed spaces:

$$\langle x, y \rangle = \frac{1}{4}\left(\Vert x+y \Vert^2 - \Vert x - y \Vert^2\right).$$