Definition:Simple Function/Banach Space
Jump to navigation
Jump to search
Definition
Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ be a Banach space over $\GF$.
Let $f : I \to X$ be a function.
We say that $f$ is simple if and only if there exists:
- Lebesgue measurable subsets $\Omega_1, \ldots, \Omega_r$ of $I$ with finite Lebesgue measure
- $x_1, \ldots, x_r \in X$
such that:
- $\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$
for each $t \in I$.
Sources
- 2011: Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber and Frank Neubrander: Vector-valued Laplace Transforms and Cauchy Problems (2nd ed.) ... (previous) ... (next): $1.1$: The Bochner Integral