Action of k* on a variety induces grading? Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ homogeneous of degree $d$ if for all $v\in V$ and all $\lambda\in\Bbbk^\times$, we have $f(\lambda.v)=\lambda^d f(v)$. 
My question is: Does this define a grading on $\Bbbk[V]$? 
I was convinced that it is true, but I am running into difficulties. Let us first assume $\Bbbk=\mathbb{C}$, the ground field should not be an obstruction. The linear span of $\Bbbk^\times f$ decomposes since $\Bbbk^\times$ is reductive, but I don't see how to turn this into a grading on all of $\Bbbk[V]$. 
If it is true, I would really like to see a proof - it should use as little machinery as possible.
 A: Turning the action map of varieties into a map of rings, we get a ring map $\phi$ from $k[V]$ to $k[V][t,t^{-1}] $, the coordinate ring with an extra invertible variable (the coordinate on $k^*$) adjoined.  Now, for any function $\phi(f)=\sum_{i\in \mathbb{Z}}f_it^i$ for some $f_i$'s, almost all of which are 0.  Note that $f=\sum f_i$, which we obtain by restricting the function to $t=1$. Using associativity, applying $\phi$ again to the $f_i$'s is the same as applying pull-back by the multiplication map to t.  Thus, as functions on $V\times k^*\times k^*$ (letting $t,u$ be the two coordinates)
$$\sum_{i\in \mathbb{Z}}\phi(f_i)u^i=\sum_{i\in \mathbb{Z}} f_i t^i u^i$$ 
since the pull-back of the coordinate by multiplication is just the product of the coordinates .  Thus, $\phi(f_i)=f_it^i$.  
We can define the grading by letting $f$ be homogeneous of degree $i$ if $\phi(f)=ft^i$.  We have already seen that every element can be written uniquely as a sum of such elements (the $f_i$'s), and this is multiplicative since $\phi$ is a ring homomorphism.
Alternatively, we can note that we have proven that the span of the $f_i$'s is an finite-dimensional invariant subspace containing $f$, so we can apply your argument.  In general, essentially the same argument shows that the action of any affine algebraic group on the coordinate ring of any affine variety by pull-back is a locally finite action: any function is contained in a finite-dimensional invariant subspace.
