Maximal monotone operators - Zeidler's book - question about the proof.

Question

I have a problem with understanding the proof of the proposition 32.10 which I enclose below (E. Zeidler II B). I cannot understand why Zeidler gets from (28)

$$\int_{0}^{T}\varphi'(t)v(t)+\varphi(t)w(t)dt=0$$ for all $\varphi \in C_{0}^{\infty}(0,T)$?

Answer 1

Think about it as a linear functional, i.e., a member of $V^{*}$, the action of which on every element of $V$ gives zero. Such a linear functional must then be zero itself.