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?