I have just read that $\{1, x, x^2, x^3, ...\}$ is a basis for $L^2(-1, 1)$...I don't get this. It seems like that set is a basis for the space of polynomials of degree up to infinity. I don't see how it can be a basis for all square integrable functions?

  • $\begingroup$ Technical point: Is the interval closed or open? $\endgroup$ – Andrew D. Hwang Nov 15 '15 at 16:30
  • $\begingroup$ As stated where I read it, it is open. $\endgroup$ – Riggs Nov 15 '15 at 17:14

In the setting of normed linear spaces, a basis spans a dense subspace. To say the monomials constitute a basis of $L^{2}[-1, 1]$ amounts to the assertion that for every function $f$ in $L^{2}[-1, 1]$, there exists a sequence $(p_{n})$ of polynomials such that $$ \int_{-1}^{1} |f - p_{n}|^{2} \to 0. $$

The space of continuous functions on $[-1, 1]$ is dense (with respect to the $L^{2}$ norm) in $L^{2}[-1, 1]$ by Lusin's theorem, and the space of polynomials is dense with respect to the sup norm by the Weierstrass approximation theorem, hence dense with respect to the $L^{2}$ norm on $[-1, 1]$. A dense subspace of a dense subspace is a dense subspace, so the monomials are indeed a basis of $L^{2}[-1, 1]$.

If you did mean $L^{2}(-1, 1)$ (open interval), you can use the fact that the space of compactly supported functions in $L^{2}(-1, 1)$ is dense, then use the sketch of the preceding paragraph to see that polynomials are dense in the space of compactly supported $L^{2}$ functions.


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