# Doubts in Proposition 3.14 in Haim Brezis

I was reading Haim Brezis Weak* topology section on page 6 4

I understand everything except fact that

why in particular $$||=0\implies \phi(f)=0$$

Any Help in this regard will be appreciated Thanks you

If $$\langle f,x_i\rangle=0$$ for all $$i$$, then $$tf\in V$$ for all $$t\in \mathbb R$$. This implies $$\phi(tf) = t \phi(f) <1 \quad \forall t\in \mathbb R,$$ and $$\phi(f)$$ has to be zero.