History of Banach Spaces and Linear Operators

Other subjects such as topologies, measures and integrals, locally convex linear spaces, Banach lattices, and Banach algebras are treated only in so far as they turn out to be relevant for this purpose. Results about non-self-adjoint operators on Hilbert spaces have been a source of inspiration for the theory of operators on Banach spaces.

---

make it look much more logical, but actually it happens quite differently. • Atiyah [2004 ] The monographs A. F. Monna: Functional Analysis in Historical Perspective (1973), and J. Dieudonne: ´ History of Functional Analysis (1981), as well as all articles devoted to the history of functional analysis deal only with the development before 1950. Now the time has come to cover the second half of the twentieth century too. I have undertaken this adventure. Let me introduce myself by telling you that I received my M. Sc. degree in 1958, just at the time when the renaissance of Banach space theory started. Thus I have ?rst-hand experience of the progress achieved during the past 50 years. Due to the explosion of knowledge, writing about functional analysis as a whole seems to be no longer possible. Hence this book is focused on Banach spaces and (abstract bounded) linear operators. Other subjects such as topologies, measures and integrals, locally convex linear spaces, Banach lattices, and Banach algebras are treated only in so far as they turn out to be relevant for this purpose. The interplay with set theory is described carefully: Which axioms are needed in order to prove the Hahn–Banach theorem? Results about non-self-adjoint operators on Hilbert spaces have been a source of inspiration for the theory of operators on Banach spaces. Such topics are discussed in great detail. However, I have omitted almost all operator-theoretic considerations that depend decisively on the existence of an inner product.