Does there exist a generalization of distributions?

Question

We know that distributions are generalizations of mappings from $\mathbb{R}^n$ into $\mathbb{C}$. As an analog, is there a generalizations of mappings $\mathbb{R}^n$ into $X$ where $X$ is assumed to be a Banach space. Moreover, does a similar generalized Fourier transform exist? In the analog sense of mapping tempered distributions onto tempered distributions.

Answer 1

Yes, and it is very useful for PDE. A lot of evolution problems can be written under the form $$u^\prime(t) = Au(t)$$ where $u$ is a distribution on $\mathbb R$ with values in a Banach or Hilbert space $E$ (typically a Sobolev space).

For more details, you can check for instance the chapter 39 "Functions and Distributions Valued in Banach Spaces" of Trèves, François. Basic linear partial differential equations. Vol. 62. Academic press, 1975.