Although Function Spaces play an important role already
for a long time, be it the family of $L^p$-spaces in
classical Fourier analysis, the Hardy-spaces for
the treatment of Calderon-Zygmund operators or
(anisotropic) Besov spaces in micro-local analysis,
there is still only a small family (usually $L^p$-spaces
and Besov spaces) which are really in regular use by the
majority of mathematicians working in analysis.
The talk will try to outline some genera construction
principles for function spaces (rather than concrete,
multi-parameter spaces), meaning typically Banach spaces
of distributions, the role of the theory of Banach frames,
description by atomic decompositions and corresponding
invariance properties. Group representation theory
is playing an important role in the description(and analysis)
of these function spaces, in particular if one takes the
approach via coorbit spaces.
The talk will be more in the spirit of motivation, the
discussion of general principles and the transfer of ideas
in one setting rather than the detailed technical presentation
of a few special cases.