Just out of curiosity, are $L^p$ spaces connected? Does the connectedness depend on $p$?
2026-03-29 01:33:54.1774748034
Are $L^p$ spaces connected?
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Any element $x$ is path-connected to the zero vector by a path $\gamma_x:[0,1]\to L^p, \gamma_x(t)=tx$, and this is a property that holds in any normed vector space, not just $L^p$.