It is well-known that the Cauchy problem for the heat equation \begin{align} &\partial_t u-\Delta u =f\\ &u|_{t=0}=u_0, \ \text{and} \ f \in\mathcal{S}' \end{align} has a unique solution in $C([0,+\infty);\mathcal{S}'(\mathbb{R}^d))$, where $\mathcal{S}'(\mathbb{R}^d)$ is the class of tempered distributions.
How can we prove this?