Let $L$ be a continuous linear functional on a Banach space $X$, put $Y=\ker L$. Suppose there is a nonzero $x_0$ in $X$ such that $\|x_0+y\|\ge\|x_0\|$ for all $y\in Y$. Prove that $\|L\|=|L(x)|$ for some $x$ in the closed unit ball.
This was an old functional analysis qualifying exam problem. The original statement asked to prove an iff statement, and the converse was easy. This direction however I'm pretty clueless about. Does anybody have a hint?
Let $x := \frac{x_0}{\|x_0\|}$. Using the fact that, by assumption: $1 = \|x\| = \min_{y \in \ker(L)} \|x + y\|$ (by dividing the $y$s in the assumption by $\|x_0\|$ and relabeling them), and that in general $\inf_{y \in \ker(L)} \|x + y\| = \frac{|L(x)|}{\|L\|}$, we obtain: $1 = \frac{|L(x)|}{\|L\|}$, hence: $|L(x)| = \|L\|$.