Theorem : Dunford-Pettis Theorem

Question

I don't understand why : there exists a function $g_y(.)\in L^{\infty}(0,T)$ with : $$ \langle Bf,y\rangle=\int_{0}^{T}{f(t)g_y(t)dt}.\qquad (12.2.7) $$

$$ \|g_y\|_{L^{\infty}(0,T)}\leq\|B\|_{(L^{1}(0,T),X^{*})}\|y\|_{X}\qquad (12.2.8) $$

Answer 1

This is immediate from the fact that dual of $L^{1}$ is $L^{\infty}$. The norm of $g_y$ you get from this duality is $ \leq \sup \{|\langle Bf , y \rangle|: \|f\|_1 \le1 \} \leq \|B\|\|y\|_X$