The subject of classical harmonic analysis is to study Lebesgue integrable functions and complex-valued regular measures on the compact abelian group \(\mathbb T = \mathbb R/2\pi\mathbb Z\). The corresponding spaces \(L^1(\mathbb T)\) and \(M (\mathbb T)\) are Banach spaces and an important tool is the Fourier transform.
The author surveys some results in harmonic analysis from the point of view of Banach space theory. Let us give an example. The Fourier transform is a continuous, linear, and one-to-one operator from \(L^1(\mathbb T)\) into \(c_0(\mathbb Z)\). Is it onto? The answer is no and the author gives two different proofs. On the one hand, he gives a concrete example of an element of \(c_0(\mathbb Z)\) that does not belong to the image of the Fourier transform, on the other hand, he uses the fact that \(L^1(\mathbb T)\) is weakly sequentially complete and \(c_0(\mathbb Z)\) not.