Do you help me to:
checking that a linear map $u : X \longrightarrow Y$ between Banach spaces is not bounded below if and only if there is a sequence of unit vector $(x_{n})$ in $X$ such that $\lim_{n\rightarrow \infty} u(x_{n})=0$
If $u$ is not bounded below then for each $n$ there exists $z_n$ such that $\| u(z_n)\|<\frac{1}{n}\|z_n\|$. In particular $\|z_n\|\neq 0$. Now define $x_n=\frac{z_n}{\|z_n\|}$. Then $\{x_n\}$ is a sequence of unit vectors such that $\lim_{n\rightarrow\infty}u(x_n)=0$. The other implication is easy.