I need to solve an exercise that asks for the vector associated to a certain form of the Hilbert space $L^2(\mathbb{R})$. I am pretty sure this "associated vector" is related to Riesz's representation theorem, but I am not sure how to proceed. The form is given by:
$$F(f) = \int_0^1 f(x^2) x dx$$
While it is easy to check that it is linear:
$$F(\lambda f + \mu g) = \int_0^1 (\lambda f + \mu g)(x^2) x dx = \lambda \int_0^1 f(x^2) x dx + \mu \int_0^1 g(x^2) x dx = \lambda F(f) + \mu F(g)$$
I don't think it is continuous, because I can't find a $g\in L^2(\mathbb{R})$ so that $F(f) = \langle f | g \rangle = \int_{-\infty}^{+\infty} f(x)g(x)dx$. I may be confused, but I think that since we are in $L^2(\mathbb{R})$, we should use the usual inner product and I don't see a way to equate that with the definition of $F$, which has an x^2 as the argumento of the function in the integrand. Any help would be appreciated. Thanks.