Dominated convergence theorem vs continuity

Question

Let $\{f_n\}$ be a sequence of functions in $L^2(0,1)$ such that $\lim_n f_n = f$ pointwise and $\vert f_n(x) \vert \leq g(x)$ for some integrable function $g$. By the dominated convergence theorem it follows that $$ \lim_n \int f_n = \int f. $$ However, $L^2(0,1)$ is equipped with the inner product $$ (f,g) = \int f\bar{g} $$ which in particular is continuous in its first argument (one realizes this since $T_g(f) = (f,g)$ is a bounded linear operator.) Hence for any sequence $f_n$ converging to $f$ (in the $L^2$ norm) we have that $$ \lim_n T_g(f_n) = T_g(f) \iff \lim_n \int f_n \bar{g} = \int f \bar{g}. $$ Putting $g \equiv 1$, we get the same conclusion one gets from the dominated convergence theorem, however with weaker assumptions being put on our sequence $\{f_n\}$. Indeed instead of pointwise convergence, we only require convergence in norm; also the requirement of being dominated is dropped completely. So it seems to me that the dominated convergence theorem is of little need in a finite space? Is this reasoning correct or am I mixing some concepts up?