Does there exist a $\sigma$-compact complete metric space which is not locally compact? [closed]

Question

Give an example of a $\sigma$-compact complete metric space which is not locally compact. A space $X$ is said to be locally compact if for each $x\in X$, there exist an open set $U$ and a compact subset $K$ of $X$ such that $x\in U\subseteq K$.

Answer 1

Let $\mathcal H$ be a separable Hilbert space with basis $v_1,v_2,\ldots$ and let $X$ be the closed subspace of all vectors that can be written as $tv_n$ with $t\in[0,1]$ and $n\in\mathbb N$. Then $X$ is $\sigma$-compact, since it is the union of the compact sets $K_n = \{tv_n \mid t\in [0,1]\}$. But $X$ is not locally compact, because there is no compact neighbourhood of 0.