Differentiability of tempered distributions over $\mathbb{R}^3$ One of the topics I find interesting in physics is the time evolution of improper quantum states (physicists call them kets). My question to this mathematical community is:
In what exact sense is a mapping $\phi:\mathbb{R}\to \mathcal{S}'(\mathbb{R}^3)$, $t\mapsto\phi(t)$ continuous and furthermore differentiable in the variable/parameter "$t$"? ($\mathcal{S}'(\mathbb{R}^3)$ is of course the topological dual of the Schwartz test function space)
 A: Definition: $\phi : \mathbb R \to \mathcal S'(\mathbb R^3)$ is continuous if $\phi(t) \to \phi(t_0)$ whenever $t \to t_0$.
But what does $\phi(t) \to \phi(t_0)$ mean, i.e. how is convergence in $\mathcal S'(\mathbb R^3)$ defined?
Definition: $u_\lambda \to u_0$ when $\lambda \to 0$ if $\langle u_\lambda,
 \rho \rangle \to \langle u_0, \rho \rangle$ for every $\rho \in \mathcal S(\mathbb R^3).$
With $\langle u, \rho \rangle$ I mean the application of the tempered distribution $u$ on the testfunction $\rho$.

For differentiability we can just use the normal definition.
$$
\phi'(t_0) = \lim_{h \to 0} \frac{\phi(t_0+h) - \phi(t_0)}{h}$$
meaning that
$$\langle \phi'(t_0), \rho \rangle 
= \lim_{h \to 0} \frac{\langle \phi(t_0+h) - \phi(t_0), \rho \rangle}{h}
= \lim_{h \to 0} \frac{\langle \phi(t_0+h), \rho \rangle - \langle \phi(t_0), \rho \rangle}{h}
.$$
A: The topology one should use on $\mathscr{S}'(\mathbb{R}^d)$ is not the weak-$\ast$ topology but rather the strong topology. It is the locally convex topology defined by the family of seminorms
$$
||\phi||_A=\sup_{f\in A}|\phi(f)|
$$
indexed by bounded subsets $A$ of Schwartz space $\mathscr{S}(\mathbb{R}^d)$. Saying that a subset $A$ is bounded means that for all multiindex $\alpha$ and all nonnegative integer $k$,
$$
\sup_{f\in A}\sup_{x\in\mathbb{R}^d}\langle x\rangle^k|\partial^{\alpha}f(x)|\ <\ \infty\ .
$$
Now continuity of $t\mapsto \phi(t)$ at $t_0$ means that for all bounded set $A$,
$$
\lim_{t\rightarrow t_0}||\phi(t)-\phi(t_0)||_A=0\ .
$$
Finally, differentiability at $t_0$ with derivative equal to some distribution $\psi$ in $\mathscr{S}'(\mathbb{R}^d)$ means that for all bounded $A$,
$$
\lim_{t\rightarrow t_0}||(t-t_0)^{-1}[\phi(t)-\phi(t_0)]-\psi||_A=0\ .
$$
