Infinite dimensional $\sigma$-compact space

Question

It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.

Answer 1

Consider the subspace $V_\infty$ of $\mathscr l^2(\mathbb N)$ where only a finite amount of terms in a series is non-zero. This is an infinite dimensional normed vector space.

Define also the subspace $V_n$ where only the first $n$ terms of a series are non-zero. $V_n \cong \mathbb R ^n$ with the standard norm. As such there is a sequence of compacta $K_{n,i} \subset V_N$ so that $\bigcup_{i\in \mathbb N} K_{n,i} = V_n$. The $K_{n,i}$ are compact in $V_n$, but since $V_n$ has the subspace topology they are also compact in $V_\infty$.

Since every $x\in V_\infty$ lies in one of the $V_n$, and clearly $V_n\subset V_\infty$ for all n, we have $\bigcup_{n\in \mathbb N} V_n = V_\infty$. Putting this together with the previous line:

$$V_\infty = \bigcup_{n \in \mathbb N}V_n = \bigcup_{n \in \mathbb N}\bigcup_{i\in \mathbb N} K_{n,i} = \bigcup_{(n,i) \in \mathbb N\times \mathbb N} K_{n,i}$$

So $V_\infty$ is $\sigma$-compact.