Let $$T be a non-zero linear functional from a normed linear space $X$ to $K$($\mathbb{R}$ or $\mathbb{C}$). Prove that $T$ maps open sets to open sets.
My Attempt: Since $T$ is a non-zero linear functional it is onto. Now if I can prove that there exists a $\delta>0$ s.t. for every $y\in K$ there exists $x\in X$ with $\|x\|<\delta \|y\|$ and $Tx=y$, then we are done. But I am not getting how to do it. Thanks in advance.
Let $\mathrm{H}$ be the null space of $T$ and $a \notin \mathrm{H}.$ Then, $\mathrm{X}$ is the (algebraic) direct sum of $\mathrm{H}$ and $a \Bbb K.$ If $\mathrm{A}$ is open and $z \in \mathrm{A}$ is a point in $T^{-1}(\{\lambda\})$ (that is, $Tz = \lambda$ and I will construct an interval surrounding $\lambda$), there exists a ball $\mathrm{B}(z)$ with centre $z$ and some radius $\eta > 0.$ Then, $$\begin{align} T(\mathrm{B}(z)) &= T(\mathrm{B}(z) \cap (z + a \Bbb K)) \cup T(\mathrm{B}(z) \cap (z + \mathrm{H})) \\ &= T(z) + T(\mathrm{B}(0) \cap a \Bbb K) \cup T(\mathrm{B}(0) \cap \mathrm{H}) \\ &= \lambda + (-\eta T(a), \eta T(a)) \cup \{0\} \\ &= (\lambda - \eta T(a), \lambda + \eta T(a)), \end{align}$$ which is an interval surrounding $\lambda.$ CQFD
