# Prove $\|Tx\|_\infty \le \|x\|$ for all $x$ belonging to a normed space

Here is the question:

And here is the given answer:

I would like to ask in the first line of the proof, why $$\|x^*_k\|$$ is less or equal than $$1$$? I guess zero belongs to $$B(x^*_k,r)$$ for any $$k$$, then $$\|x^*_k - 0|| but I don't know how to prove it. May I have some hints please?

• $x_k^{*} \in B_E^{*}$ so $\|x_k^{*}\| \leq 1$. That is the definition of $B_E^{*}$. May 26 at 5:48
• $E^{*}$ is the set of all bounded linear maps from $E$ to $\mathbb K$. $B_E^{*}$ is the closed unit ball of $E^{*}$. May 26 at 7:19
• @Kavi Rama Murthy Thank you. I thought that $B^*_E$ is the set of all bounded linear map from $B_E$ to $\mathbb K$ rather than $B(E^∗)$. The notations are confusing me. May 26 at 7:34