Lebegue-Bochner Spaces and Evolution Triples

Abstract

The functional-analytic approach to the solution of partial differential equations requires knowledge of the properties of spaces of functions of one or several real variables. A large class of infinite dimensional dynamical systems (evolution systems) can be modeled as an abstract differential equation defined on a suitable Banach space or on a suitable manifold therein. The advantage of such an abstract formulation lies not only on its generality but also in the insight that can be gained about the many common unifying properties that tie together apparently diverse problems. It is clear that such a study relies on the knowledge of various spaces of vector valued functions (i.e., of Banach space valued functions). For this reason some facts about vector valued functions is introduced. We introduce the various notions of measurability for such functions and then based on them we define the different integrals corresponding to them. A function $f:\Omega \to X$ is strongly measurable if and only if it is the uniform limit almost everywhere of a sequence of countable-valued, $\Sigma$-measurable functions. If X is separable and $f: \Omega\to X$, be a function then the following three properties are equivalent: (a) f is strongly measurable; (b) f is weakly measurable; (c) f is Borel measurable. A strongly measurable function $f: \Omega\to X$, is Bochner integrable if and only if the function $\omega\to \left\|f\left(\omega \right)\right\|_X$ is Lebesgue integrable (i.e., $\left\|f\left(.\right)\right\|_X\in L^{1}(\Omega)$). The emphasis of the project is on the so-called Bochner integral, which generalizes in a very natural way the classical Lebesgue integral to vector valued functions. We continue with vector valued functions and introduce the so-called Lebesgue-Bochner spaces, which extend to vector valued functions of the well known Lebesgue $L^{p}$-spaces. We also consider evolution triples and the function spaces associated with them. Evolution triples provide a suitable analytical framework for the study of a large class of linear and nonlinear evolution equations.