The modern viewpoint on modulation spaces describes them as Banach
spaces of distributions (tempered or even ultra-distributions) whose
elements are characterized by a certain decay and summability
properties of their Short time Fourier transforms with respect to
e.g. the Gaussian window. As it has been shown in many papers and in
particular in K. Groechenigs book (Foundations of TF Analysis) this
family of spaces is extremely useful when it comes to the
description of the behaviour of many important mapping arising in
Fourier analysis as well as in various application areas. The
prototype of such a mapping is the Fourier transform, which should
not be seen as just a unitary mapping on the Hilbert space $L^2$
but rather on a whole family of modulation spaces (all the ones
arising from solid function spaces on phase space which are
invariant with respect to rotation of phase space by 90 degrees!,
e.g. those arising from weighted $L^p$ spaces with radial symmetric
weights).

Since the rich variety of (new) function spaces appears a bit
confusing to new-comers it has turned out to be useful to restrict
this family to just two important case (aside from $L^2$): adding
the modulation spaces corresponding to a simple $L^1$ and its dual
corresponding to an $L^\infty$ condition (also known as
Feichtinger's Segal algebra $S_0$ and $S'_O$). The triple
$(S_0,L^2,S'_0)$ is called a Banach Gelfand triple, and the concept
of (unitary) Banach Gelfand triple morphisms is quite useful. Among
other one can claim that the Fourier transform is such an
isomorophism, uniquele determined by the fact that it maps the pure
frequencies into the corresponding Dirac measures, while the
so-called spreading mapping extends the well-known identification of
Hilbert Schmidt operators and their $L^2$ spreading symbols (over
the TF-plane) to a Gelfand triple isomorphism for a suitable Gelfand
triple of operator spaces (which have a simple description due to
the kernel theorem for operators on $S'_0$), uniquely determined by
the fact that it maps the pure TF-shift operators on the
corresponding Dirac measures (living on the TF-plane).