The exercises shows that the Riesz Representation does not hold on infinite-dimensional inner product spaces. I need help.
Suppose $C_{\mathbb{R}}([-1, 1])$ is the vector space of continuous real-valued functions on the interval $[-1, 1]$ with inner product given by \begin{equation} \langle \ f\ , \ g \ \rangle = \int_{-1}^{1} f(x) g(x) dx \end{equation} for $f, g \in C_{\mathbb{R}}([-1, 1])$. Let $\varphi$ be the linear functional on $C_{\mathbb{R}}([-1, 1])$ defined by $\varphi (f ) = f (0)$. Show that there does not exist $g \in C_{\mathbb{R}}([-1, 1])$ such that \begin{equation} \varphi(f) = \langle \ f\ , \ g \ \rangle \end{equation} for every $f \in C_{\mathbb{R}}([-1, 1])$.
If there is such a $g$, then $\varphi$ is a continuous linear functional on the given space, and $\varphi(f) = 0$ if $f(0) = 0$. As the set of all such $f$ is (everywhere) dense in the space, this would imply $\varphi \equiv 0$.