Hahn-Banach theorem and complex linear functional

Question

I find this exercise but I cannot prove it.

If $ X $ is a complex topological vector space and $ f \colon X \to \mathbb C $ is nonzero continuous linear function, show that $ X \setminus \ker f $ is connected.

Answer 1

Select $x \in X$ with $f(x) = 1$. Observe that $f(ix) = i$ so that $ix \notin \ker f$ and that $\phi(t) = tx + (1-t)ix$ is a path in $X \setminus \ker f$ from $x$ to $ix$.

Let $y \in X \setminus \ker f$.

• If $f(y) \in \mathbb R$ then $\psi(t) = t(ix) + (1-t)y$ is a path in $X \setminus \ker f$ from $ix$ to $y$, and the concatenation of $\phi$ with $\psi$ is a path from $x$ to $y$.
• if $f(y) \notin \mathbb R$ then $\rho(t) = tx + (1-t)y$ is a path in $X \setminus \ker f$ from $x$ to $y$.

Thus there exists a path from $x$ to $y$, implying that $X \setminus \ker f$ is path connected.