$\langle\cdot,\cdot\rangle$ is the dot product on the real vector space $\mathcal C ([0,1],\mathbb R)$ defined by $\langle f,g\rangle = \int_{-1}^1 fg$, and $(L_n)$ is the family of normalised Legendre polynomials $((X^2-1)^n)^{(n)}$ divided by their $\langle\cdot,\cdot\rangle$-norm. Given a function $f$ such that the series $$ \sum \langle f,L_n \rangle $$ converges absolutely, I have to show that the series of functions $$ \sum \langle f,L_n \rangle L_n $$ converges pointwise to $f$.
Now, I know that it converges to $f$ in the $\mathcal L^2$ sense: $\sum_{k=0}^n\langle f,L_n\rangle L_n \xrightarrow{\|\cdot\|_2}f$ since the $(L_n)$ form a complete orthonormal set of vectors, and I would only need to show the aforementioned series converges pointwise to some function $g$ to conclude, as then $f-g$ would be of ${\|\cdot\|}_2$-norm zero and thus zero. How do I show this ?