Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties

In this note we prove the Banach space properties of the homogeneous Newton-Sobolev spaces $H{N}^{1,p}(X)$ of functions on an unbounded metric measure space $X$ equipped with a doubling measure supporting a $p$-Poincaré inequality, and show that when $1<p<\mathrm{\infty }$, even with the lack of global $L^p$-integrability of functions in $H{N}^{1,p}(X)$, we have that every bounded sequence in $H{N}^{1,p}(X)$ has a strongly convergent convex-combination subsequence. The analogous properties for the inhomogeneous Newton-Sobolev classes $N^{1,p}(X)$ are proven elsewhere in existing literature