let $V$ be an inner product space. Let $X$ a subspace of $V$ and $X'$ its orthogonal complement i.e, $V=X\oplus X'$. Let $G$ be a group $G$ acting on $V$.
an element in $X\oplus X'$ is it a couple $(x,x')$ or a sum $x+x'$, i'm asking because the map $V\times V \rightarrow V; \,(x,x')\mapsto x+x'$ is not injective
if each $v\in V$ is written $v=x+x'$ so how is the action corresponding to this decomposition? can we write $g(x+x')=gx+gx'$
supppose $G$ fixes all elements of $X$, can we write $V/G$ is homeomorphic to $X\times (X'/G)$?