Let $$ L^2:=\left\{f:[0,1] \to \mathbb R \; \middle| \; \int_0^1 |f|^2 < \infty\right\}, $$ $$ \langle f,g \rangle:= \int_0^1 f g, $$ and $$ \|f\| := \sqrt{\langle f, f \rangle}. $$
Prove that $\|f_n-f\| \to 0$ as $n\to\infty$ exactly when the limit of $\langle f_n, g_m \rangle$ as $n,m \to \infty$ exists. Can someone give me a hint? What bugs me is that $f_n$ converges but $g_m$ doesn't necessarily converge. Let's say $f_n$ and $g_m$ converge. Then I would get the following: $$ \langle f_n – f + f, g_m – g + g \rangle = \langle f_n – f, g_m – g \rangle + \langle f_n – f, g \rangle + \langle f, g_m – g \rangle + \langle f, g \rangle, $$ Which converges to $\langle f, g \rangle$ as $n, m \to \infty$.
Since $\|f_n - f\| \to 0$ as $n\to\infty$, we know $f_n \to f$ in $L^2[0,1]$, and hence $(f_n)$ is a Cauchy sequence: $$ \|f_n - f_m\| \to 0 $$ as $n,m \to \infty$, speaking loosely. More precisely, for any $\varepsilon > 0$, there's an $N \in \mathbb N$ such that $n, m \geq N$ implies \begin{align*} \varepsilon &\geq \|f_n - f_m\| && \mbox{Def'n of Cauchy seq.} \\ &= \| f_n - g_m + g_m - f_m \| \\ &= \|(f_n - g_m) - (f_m - g_m) \| \\ &\geq \bigg| \| f_n - g_m \| - \| f_m - g_m \| \bigg| && \mbox{Reverse triangle inequal.} \\ \end{align*}
Of course, as soon as we stuck $g_m$ inside the norm, we made some assumptions about it, at least $g_m \in L^2[0,1]$ (related to assumptions made by your statement about $\langle f_n, g_m \rangle$). Does this help?