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

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?

• Simply consider the composition $X \to X/R \cong \mathbb{K}$. It has kernel $R$ and is continuous as a composition of continuous functions. Commented Jun 13, 2021 at 17:25

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.