Consider the following hyperplane separation theorem: Suppose $E$ is a proper convex subset in a normed vector space $\mathcal{X}$ and $\theta$ (the origin) is an interior point of $E$. And suppose $x_0\notin E$. Then there exists a hyperplane denoted by $H_f^r$ that separates $x_0$ and $E$. Here $H_f^r=\{x| f(x)=r\}$ where $f$ is a continuous and linear functional and $r\in\mathbb{R}$.
I have learned in the book Functional Analysis (2ed) written in Chinese by the professor Gong-Qing Zhang from the Peking University (shown in the figure below) that the condition of nonempty interior is necessary when $\mathcal{X}$ is of infinite dimensions. How can we show this by an example? That is to say I hope to construct a counterexample of the above separation theorem if $E$'s interior is empty.
The text with blue underline in the above figure means in English:
But, for a infinite-dimensional normed vector space, the condition of nonempyty interior cannot be omitted.
Moreover, can the condition of nonempty interior be removed in the case of finite dimensions? If so, how can we prove it?
Any answers will be appreciated.