ABSTRACT
In this chapter, the authors start with normed spaces, while they concentrate on Hilbert spaces. They consider integration theory for measurable functions assuming their values in a Banach space. The authors recalls various forms of the Hahn–Banach theorem and also considers a self-improvement of the triangle inequality. The Komlos theorem is a tool to create the limit when the authors have a bounded sequence in a normed space. The Bochner integral is an advanced topic for beginners. In a word the Bochner integral replaces R or C in the range by Banach spaces. The authors extend the notion of measurability of functions to Banach spaces-valued functions. They then consider some convergence theorems in analogy with the ones in the theory of the Lebesgue integral. The notion of weak measurablity corresponds to the separation of the real-valued functions into the positive part and the negative part.
