Having $X$ a normed vector space. If $f$ is a linear operator from $X$ to $ℝ$ and is not continuous in $0$ (element of $X$) , how can we show that there exists a sequence $x_n$ that converges to $0$ for which we have $f(x_n) = 1$ (for all $n$ element of $ℕ$).
Any help would be greatly appreciated, thank you.
By definition of a limit, construct a sequence $x_n$ such that $x_n\rightarrow 0$ but $\|f(x_n)\|\geq\delta>0$. Here we use $f(0)=0$. Then rescale by letting $$u_n=\dfrac{x_n}{\|f(x_n)\|}$$ Clearly $\|f(u_n)\|=1$ by linearity of $f$ but $u_n\rightarrow 0$ since $x_n$ does and $\|f(x_n)\|\geq\delta$.