Decomposition of the field of Laurent formal power series in a polynomial ring and a formal power series ring

Question

Can we decompose the field of formal Laurent series as $$\mathbb C((t))\cong \mathbb C[t^{-1}]\times\mathbb C[[t]]$$ as vector spaces over the field of complex numbers? The map

$$\sum_{i\in\mathbb Z}\lambda_it^i\mapsto (\sum_{i<0}\lambda_it^i,\sum_{i\geq0}\lambda_it^i)$$

seems to satisfy the above decomposition.

Answer 1

Yes, the map that you defined is an isomorphism of vector spaces. (Obviously not an isomorphism of algebras, but you knew that already.)