Banach Gelfand triples are triples of Banach spaces included in each other in a very special way: Formally one assumes that there is an embedding of a Banach space (of test functions) into its dual (a Banach space of generalized functions), with a Hilbert space in the middle.

There is a specific Banach Gelfand triple, based on the Segal Algebra S_0(R^d) (in fact it can be defined for general LCA groups), which has its roots in time-frequency analysis.
In fact, this space (and its dual as well as the intermediate Hilbert space L^2(R^d)) are so-called modulation spaces and can be characterized via their Gabor coefficients.

This BGT is suitable in describing the Fourier transform in a general way, but also the transition from operators to their spreading representation. One of the cornerstones is the analog of matrix representations for linear mappings on R^n:
the so-called kernel theorem, where one usually has to resort to the Schwartz space of rapidly descreasing function, a prototype of a nuclear Frechet space (hence not a Banach space!).