Let $T \in \mathscr D^\prime(\mathbb R)$ be a distribution and let $\delta_0$ be the usual Dirac mass at $0$ (in $\mathbb R$).
Let us define the functional $S:=T\otimes \delta_0$ in the following way: for every $\phi \in C_c^\infty(\mathbb R^2)$ we set $$ \langle S, \phi \rangle := \langle T, \phi(\cdot, 0) \rangle, $$ which makes sense being $\phi(\cdot, 0) \in C_c^\infty(\mathbb R)$.
Question. Is $S$ a distribution?
I do think the answer is yes, because the map $S$ is clearly linear, and continuity should also be true (?). Is this an instance of the so-called tensor product of distributions? To which distributions can this construction be generalized?
I have always heard that defining product of distributions is "hard" (apart in the trivial case, i.e. when one distribution is represented by a smooth compactly supported function).