Prove a property of the weight decomposition of representation of $\mathfrak{sl}(2,\mathbb{C})$ I try to prove the following property of complete decomposition of the representation of $\mathfrak{sl}(2,\mathbb{C})$:

Show if $V$ is a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb{C})$, then 
  $$V\cong \bigoplus n_k V_k$$
  and $n_k=\dim V[k]-\dim V[k+2]$.

The last part (about $n_k$ and the dimension of the eigenspace) seems to be a very amazing result for me. 
Since $V_k$ does not contain $V[k+2]$. And we can take any numbers of copies of $V_k$. So I am really confused here since I think these two numbers at each side are totally irrelevant. So I must miss something very crucial. Please point it out. Thanks!

I assume that $V_k$ are the standard irrep of weight $k$. Then, consider multiplication by $x : V[k] \to V[k+2]$. Clearly, $x$ is surjective and moreover, $x(v) = 0$ iff $v$ is a heighest weight vector of weight $k$. So $\dim \ker(x) = \dim V[k] - \dim V[k+2]$ is indeed $n_k$.
Here is an intuitive explanation : there are two kind of eigenvector of weight $k$, the one which spans an irrep $V_k$ and the one which comes from irrep of bigger weight. For example, if $w \in V_{k+2}$, then $y(w) \in V_k$ is a vector of "second kind". Now, by definition, the "second kind" vectors are exactly the vectors which are not killed when multiplied by $x$. So $\dim \ker x$ is the number you were looking for, and it is $\dim V[k] - \dim V[k+2]$. 
