Suppose we have the PDE
$\frac{\partial u}{\partial t} - \Delta u = f(x,t)$
with some boundary conditions.
I am confused about what it means to say that the weak time derivative $u_t \in L^2([O,T];H^{-1})$. It makes no sense to me. My teacher has written:
We have $u_t = \Delta u + f(x,t)$ so $u_t \in L^2([O,T];H^{-1})$.
As far as I'm concerned, $u_t$ for a fixed $t$ is a function, not a functional that takes in a $H^1_0$ function and gives out a real number. What am I doing wrong?