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)?