I have a solution, $u(x,t)$, to the heat equation $u_t-u_{xx} = 0$ with initial condition $u(x,0) = f(x)$.
I would like to show that as $t \rightarrow 0$, the solution $u(x,t)$ tends to the initial condition $f(x)$. In other words
$$ \lim_{t \rightarrow 0} \int|u(x,t) - f(x)|^2 = 0 $$
My solution $u(x,t)$ is an infinite series. I think I need to use the M-test to show that $u(x,t)$ converges uniformly to $f(x)$, but i'm not sure where the fact that this convergence is in $L^2$ comes to play.
Thanks