The purpose of this talk is to give a historical perspective on some aspects of the theory of function spaces, i.e. Banach spaces of functions (or distributions, when one looks at the dual spaces).

The original approach to Besov spaces (Besov, Taibleson, Stein) came from the idea of generalized smoothness, expressed by (higher order) difference expression and the corresponding moduli of continuity, e.g. expressing smoothness by the decay of the modulus of continuity (via the membership in certain weighted Lq-spaces on (0,1]). Alternatively there is the line described in the book of S.Nikolksii characterizing smoothness (equivalently) by the degree of approximation using band-limited functions.

The second age is characterized by the Paley-Littlewood characterizations of Besov or Triebel-Lizorkin spaces using dyadic decompositions on the Fourier transform side, as used in the work of J.Peetre and H.Triebel, the masters of interpolation theory. Their contribution was to show that these families of function spaces are stable under (real and complex) interpolation methods.

The third age is - from our point of view - the characterization of function spaces in the context of coorbit spaces, using irreducible integrable group representations of locally compact groups.

Let us also remind that the concept of retracts plays an important role in the context of interpolation theory (see the book of Bergh-Loefstroem), and can be used to characterize Banach frames and Riesz projection bases.