If one is dealing with sets, given a function $f:A \to B$, its inverse is a function $g:B \to A$ such that $f \circ g= id_B$ and $g \circ f= id_A$. This carries over to linear functions defined on finite dimensional vector spaces, too. But let's take a continuous linear operator from a Banach space to another,for instance the Fourier transform $\mathcal{F}$. It is known that $\mathcal{F}$ maps $L^2 (\mathbb{R})$ into $L^2 (\mathbb{R})$ (thanks to Plancherel's theorem) and it is trivial to verify its boundedness from $L^1$ to $L^\infty$. Thanks to interpolation, one can thus say that $\mathcal{F}: L^p \to L^q$ continuously with $p \in [1,2]$ and $q \in [2, + \infty]$ its Holder conjugate.
To define the Fourier transform "directly", one should need $f \in L^1$, but thanks to continuity I should be able to work on a dense and "nice" subset of $L^p$ where everything works and then define the Fourier transform of a function $f \in L^p$ approximating $f$ with "nice" function, defining the $\mathcal{F}$ as a limit.
The class of "nice" dense function that I am going to take is the Schwartz functions, that is "rapidly decreasing" smooth functions $\mathcal{S}(\mathbb{R})$. It can be proved that $\mathcal{F}: \mathcal{S} \to \mathcal{S}$ continuously, and that $\mathcal{S}$ is dense in $L^p$ with $p \in [1, \infty)$. Now, if I understood well, you can define the Fourier transform on every $L^p$ with $1\leq p< \infty$, but you have to take the codomain in a way that gives you continuity of the operator. So, to sum up, given $f \in L^p$ with $p \in [1,2]$:
$$\mathcal{F}(f)= \lim_n \mathcal{F}(f_n) \; \text{ that converge in } L^q$$
with $f_n \in \mathcal{S}$. If $p >2$, theory of distributions comes up, and I should only have that $\mathcal{F}: L^p \to \mathcal{S}'$.
Let's say that I want to invert the Fourier transform. If I work with $\mathcal{S}$ everything works fine, but if I consider $\mathcal{F}: L^p \to L^q$, (with $p\in [1,2]$ and $q \in [2, \infty]$) using the classical insiemistic definition of "inverse" would lead me to search for an operator from $L^q$ to $L^p$, and this shouldn't work, because $\mathcal{F}^{-1}$ should be well defined only if its codomain is the tempered distributions, that is not every function in $L^q$ has an inverse Fourier transform that is still a function. So, what is a formal argument that puts everything in order? I feel like one should use a density argument, maybe restricting to the image of the Fourier transform, but it somehow still puzzles me a bit that if the inverse of a function from $A$ to $B$ is not a function fromm $B$ to $A$