$L^p$ spaces are always connected?

Question

Let $\mu$ be a Borel measure on $\mathbb{R}^d$. Is it possible for $L^p_{\mu}(\mathbb{R}^d)$ to be disconnected?

Sketch of Proof I assume not since I've heard a folklore that all Banach spaces are homoeomorphic and $L^2(\mathbb{R})$ (Lebesgue measure) is contractible.

Maybe I'm wrong since I'm thinking of $\sigma$-finite measures...

Answer 1

Every topological vector space (and thus every normed space) is path-connected, and thus connected.

Indeed, let $V$ be a topological vector space (or simply normed space if you want it less general). Then given $v,w \in V$, the map

$$[0,1] \to V: t \mapsto tv+(1-t)w$$ is clearly continuous, showing that $V$ is path-connected.

In particular, $L^p_\mu$ is connected so the answer to your question is always negative, for any measure $\mu$.

Answer 2

All topological vector spaces $V$ are contractible, in particular path connected. A contraction is given by $$H : V \times [0,1] \to V, H(v,t) = tv.$$

However, it is not true that all Banach spaces are homoeomorphic. For example, $\mathbb R^n$ and $\mathbb R^m$ are hoeomorphic iff $n = m$.