In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", Prop 1.10):

Let $G$ be the affine algebraic group, $e_1,e_2,\ldots$ be a basis of its regular function ring (possibly infinite). Then for any regular function $f$, we can write $f(gh)=\sum_{i=1}^M v_i(g)e_i(h)$. (Borel wrote this as tensors but the idea is the same.) Thus the orbit of $f$ under the right action of $G$ lies inside the finite dimensional space $V$ spanned by the $v_i$'s, and the orbit gives a finite dimensional subspace $W$ which is stable under the action of $G$. (Then we proceed with this finite dimensional invariant $W$ into some $GL_N$)

My question is: Is there an example that $V\ne W$? Or can we prove $V=W$ under some conditions? I only check the affine line case and could not beat $V=W$.

Thank you!