3
$\begingroup$

Let $V$ be the polynomials space over $R$ of degree less than 3 with inner product

$$\langle f,g\rangle= \int^{1}_{0} f(x)g(x) dx $$

if $x \in R$, compute $g_{x}$ such that $\langle f,g_{x}\rangle = f(x)$ for all $x$.

i suppose $f = a_{0} + a_{1}x + a_{2}x^{2} + a_{1}x^{3}$ and $g_{t}= b_{0} + b_{1}x + b_{2}x^{2} + b_{1}x^{3}$ and use $\langle f,g_{x}\rangle = f(x)$

but these calculations are to mechanical and long, is there ant theorem about functional and his representation by inner product that give the direct result?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.