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?


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