is the space of finite sequences Frechet?

Question

Let $E:=\coprod_{\mathbb{N}}\mathbb{R}$ be the space of all finite, real sequences equipped with the final structure wrt to all injections $inj_k(x)=(0,0,\dots,x,0,\dots)$. Since completeness of $E$ is inherited by completeness in $\mathbb{R}$, how do I see that $E$ has (no) countable subbasis of seminorms?

Answer 1

Then you could apply Baire's theorem so that one of the finite dimensional (closed) subspaces $E_n=\mathbb R^n \times 0\times\cdots$ would have interior points and thus coincide with $E$.