It is the purpose of this talk to discuss essentially two concepts which arose in the context of time-frequency analysis (specifically relevant for Gabor Analysis), but which represent useful and fundamental functional analytic tools which certainly will be very useful also in quite different contexts.

First of all we will discuss the meanwhile well-known concept of a (tight) frame in a Hilbert space H, and that it is equivalent to an isomorphism between the given Hilbert space and a closed subspace of l2(I). Since in a Hilbert space every closed subspace is complemented (and this fact is used in many applications) we will argue that Banach frames (typically the same families of test functions are used for large families of Banach spaces) should establish an isomorphism between a closed and complemented subspace of a suitably chosen Banach spaces of complex-valued sequences and the given Banach space. In both cases one can argue that these frames generalize the concept of a stable generating system to the context of Banach spaces.

The concept of Banach Gelfand Triples is an abstract approach to a typical situation in analysis. In order to extend an operator (such as the Fourier transform) from the Hilbert space L^2(R^d) to a larger space of generalized functions one makes use of a small space of test functions, inside of L^2, and uses the dual space for the extension. As opposed to the usual setting (of Schwartz space) we will shortly discuss the Banach algebra S_0(R^d), which is invariant under the Fourier transform as well as under time-frequency shifts. Using this specific Banach Gelfand Triple one can describe various unitary operators (such as the Fourier transform, or the mapping describing the kernel theorem) in a much better way than just with the usual setting of individual Banach spaces.