Let $X$ be an infinite-dimensional normed vector space and $\phi:X\to\mathbb{R}$ be a nonzero linear functional. If $\ker\phi$ is closed then $\phi$ is bounded.
I have some trouble to understand the solution given by the assistant:
Solution:
We have $X/\ker \phi$ is isomorphic to $\mathbb{R}$. So the map $\tilde{\phi}:X/\ker \phi\to \mathbb{R}$ is a linear map between finit-dimensional vector spaces. $\tilde{\phi}$ is bounded. So $\phi:\tilde{\phi}\circ\pi$ is bounded, where $\pi$ is the canonical projection.
Why is $X/\ker \phi$ isomorphic to $\mathbb{R}$? Thank you
