Normed linear space and non-zero linear bounded functional 
Let $V$ be a normed linear space and $l$ be a non-zero linear bounded functional on $V$.
If $d=\inf\{\|v\|:l(v)=1\}$, show that $\|l\|=\frac{1}{d}$.


You must have $\forall v \in V,|l(v)| \leq ||l||.||v||$. Without loss of generality you can assume that $l(v)=1$ (using the linearity of $l$ on the LHS and the absolute homogeneity of the norm on the RHS; of course in this case $||v|| \neq 0$). 
Your equation becomes $\forall v \in V, \frac{1}{||v||} \leq ||l||$ meaning that $||l|| = sup(\frac{1}{||v||}) = \frac{1}{inf(||v||)}$
