If $R$ is a closed subspace of a Banach space $X$ with $\text{codim } R = 1$, $\exists \psi \in X^*\backslash 0$ with $R = \ker{\psi}$

Question

Let $R$ be a closed subspace of a Banach space $X$ such that $\text{codim } R = 1$. I am trying to show that there exists $\psi \in X^*$ such that $\psi \neq 0$ and $$R = \{x \in X\mid \psi(x) = 0\}.$$ I know that it is a well-known result and that it must have been shown several times on MS but I could not find it anywhere. I was thinking that maybe it would work using the Hahn-Banach theorem but I am not able to find anything. Could any of you help me with this problem?

Answer 1

Let $Y=X/R$ (quotient space), then $\dim Y=1$ so there exists an isomorphism $$ \eta:Y\longrightarrow K $$ ($K$ is the field of coefficients), Now lift $$ X\longrightarrow Y\stackrel\eta\longrightarrow K $$ where the first map is the natural quotient map and call $\psi$ the composition.