Linearity and zeros

Question

let $f$ be a continuous lineare function defined on the banach space $E$

$f:E \rightarrow \mathbb{R}$ such that $f (x)=0$ for all $x$ such that $\|x\|<1$

so $f(x)=0$ for all $x \in E$ ( you think this is true ?)

Answer 1

For any $x$, fix $c>0$ with $c>\|x\|$. Then $$ f(x)=c\,f\left(\frac{x}{c}\right). $$ You don't need $f$ to be bounded for the argument to go through.

Answer 2

Continuity of $f$ is not necessary, but if you wish to use it, you can argue that since $f$ is zero on the open ball $B(0, 1)$, it also has to be zero on the closed ball $\overline{B}(0,1)$ by continuity.

Now you can calculate the operator norm:

$$\|f\| = \sup_{x \in \overline{B}(0,1)} |f(x)| = 0$$

We conclude that $f = 0$.