Bochner Integral: Primitive

Question

Given a Banach space $E$.

Consider a continuous function: $$F\in\mathcal{C}(\mathbb{R},E):\quad\int_\mathbb{R}\|F(s)\|\mathrm{d}s<\infty$$

Then it has a primitive: $$G(t):=\int_0^tF(s)\mathrm{d}s:\quad G'(t)=F(t)$$ How to prove this abstractly?

Answer 1

This answer is community wiki.

By Hahn-Banach one has: $$F\in\mathcal{L}(\Omega):\quad\frac{1}{\lambda(\Omega)}\int_\Omega F(s)\mathrm{d}s\in\overline{\langle F(\Omega)\rangle}$$

By continuity one has: $$F\in\mathcal{C}(\Omega,E):\quad\overline{\langle FB_\varepsilon(z)\rangle}\stackrel{\varepsilon\to0}{\to}F(z)$$

So one obtains: $$\frac{1}{h}\{G(t+h)-G(t)\}=\frac{1}{h}\int_t^{t+h}F(s)\mathrm{d}s\to F(t)$$ (This prove imitates the usual one.)