\(\displaystyle \mbox{Let}\)
. . . . .\(\displaystyle f_n,\, g_m\, \in\, L^2\,:=\,\left\{\,f\,:\, [0,\,1]\, \to\, \mathbb{R} \; \middle| \; \int_0^1\, |f|^2\, \lt\, \infty\,\right\},\)
. . . . .\(\displaystyle \langle\, f,\,g\, \rangle\,:= \,\int_0^1 \,f g\, \mbox{ for }\, f,\,g \,\in \,L^2,\, \mbox{ and}\)
. . . . .\(\displaystyle \|f\|\, :=\, \sqrt{\strut \,\langle \,f,\, f\, \rangle\,}.\)
\(\displaystyle \mbox{Prove that }\, f_n\, \mbox{ converges exactly when }\, \)\(\displaystyle \displaystyle \lim_{n,m\, \to\, \infty}\, \langle\, f_n,\, g_m \,\rangle\, \mbox{ exists.}\)
What bugs me is that \(\displaystyle f_n\) converges but \(\displaystyle g_m\) doesn't necessarily converge. Let's say \(\displaystyle f_n\) and \(\displaystyle g_m\) converge. Then I would get the following:
. . . . .\(\displaystyle \langle\, f_n \,–\, f\, +\, f,\, g_m\, –\, g\, +\, g\, \rangle\, =\, \langle\, f_n\, –\, f,\, g_m\, – \,g \,\rangle\, +\, \langle\, f_n\, –\, f,\, g\, \rangle\, +\, \langle\, f,\, g_m\, –\, g\, \rangle\, +\, \langle\, f,\, g\, \rangle\)
Then I would get:
. . . . .\(\displaystyle \displaystyle\lim_{n,m\,\to\,\infty}\,\)\(\displaystyle \langle\, f_n,\, g_m\, \rangle\, =\,\langle \,f,\, g \,\rangle\)
