Is there at least one continuous norm on $\mathbb{K}^{X}$?

Question

Let's denote by $\mathbb{K}^X$ the topological vector space of all functions from the set $X$ to base field $\mathbb{K}$ with topology generated by family of seminorms $$\{\|f\|_x = |f(x)|: x \in X\}$$ Is there at least one continuous norm on this space?I suppose that when seet $X$ is finite there always exists such norm, but how to proof that such norm never exists in case when $X$ is infinite?

Answer 1

(Assuming that $\Bbb K$ is $\Bbb R$ or $\Bbb C$.) Suppose $X$ is infinite and $||\cdot||$ is a continuous norm. Then $V=\{f:||f||<1\}$ is open, so since $0\in V$ there exist $x_1,\dots,x_n$ and $\epsilon>0$ such that if $|f(x_1)|,\dots,|f(x_n)|<\epsilon$ then $f\in V$.

Choose $f$ with $f(x_1)=\dots=f(x_n)=0$ but $f(p)=1$ for some $p\in X$ (this is possible since $X$ is infinite.) Then $$||\lambda f||<1\quad(\lambda>0),$$which implies that $||f||=0$, contradiction.