Let $V$ be an inner product space. Given any linear operator $T$ on $V$ is it always possible that $V=N(T)+R(T)$, if so how to prove it? Furthermore, is it correct to say that not every such $T$ s.t. $V=N(T)\oplus R(T)$ but if so, it uniquely determine an orthogonal projection? I'm thinking about what's needed to make a linear operator $T$ a projection, and what's needed to further make such $T$ become orthogonal projection.
I'm learning linear algebra so normally everything should be considered finite-dimensional, but the definitions of (orthogonal-)projection I read are not restricted on finite-dimensional, and it's quite confusing, for example if $V$ is infinite dimensional, and $W$ is a finite dimensional subspace of $V$, then $V=W\oplus W^\perp$, but is this still correct if $W$ is not finite dimensional?