# How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := \mbox{span }\lbrace x^2, x^3, x^4\rbrace$ Find a Hilbert basis of $U$ considering the given inner product.

I know that the Hilbert basis elements have to be orthogonal to each other, i.e. $\langle b_i, b_j\rangle = 0$ for $i\neq j$ and the elements spanning $U$ do not fulfill this criteria. I don't understand how to find this Hilbert basis where the elements are orthogonal considering the inner product and similarly make the same span as $\lbrace x^2, x^3, x^4\rbrace$.

Thank you for your help!

• Just as a note, $X$ is not a Hilbert space since it is not complete. – user38355 Dec 10 '13 at 20:55
• Hint: Use the Gram-Schmidt Process. – user38355 Dec 10 '13 at 21:05
• Thank you, I will fix this! Ok, I don't know why I never thought about that :) – user114950 Dec 10 '13 at 21:06