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In inner product spaces of polynomials, what is the point of finding the angle and distance between two polynomials? How does the distance and angle relate back to the polynomial?

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That would depend on the inner product you choose. –  tomasz Sep 8 '12 at 12:14
    
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Many spaces of functions are naturally equipped with an inner product such as $\langle f, g\rangle =\int_a^b f\bar g$. When you want to approximate an element of an inner product space by a linear combination of some simpler elements (such as low-degree polynomials) $f_1,\dots,f_n$ it helps to have the angles between $f_i,f_j$ bounded away from $0$ and $\pi$. (Ideally, all angles will be $\pi/2$, which means you have an orthogonal set of functions).

To see why, try working in $\mathbb R^2$ with the basis of $v_1=(1,0)$ and $v_2=(0.999, 0.001)$. For example, the expansion of the vector $(1,1)$ in this basis is $(1,1)=-998v_1+1000v_2$. This is just a recipe for the loss of numerical precision and all kinds of ugliness in error estimates.

Unfortunately, the "obvious" choices for basis elements such as monomials $x^n$ on the interval $[0,1]$ have small angles with respect to the aforementioned inner product. For example, the angle between $x^9$ and $x^{10}$ is about $0.05$ radian. It's better to work with Legendre polynomials instead, which are orthogonal.

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