TITLE: Are projections onto closed complemented subspaces of a topological vector space always continuous?
QUESTION [6 upvotes]: Suppose $X$ is a topological vector space and $X = V \oplus W$ is a decomposition of $X$ into closed subspaces. The decomposition gives rise to a projection $P$ onto $V$ (depending on the choice of complement $W$):
\begin{equation} P:X \rightarrow X \\v+w \rightarrow v\end{equation}
Is $P$ continuous? It seems like it ought to be (and ought to be easy to show!), but I haven't been able to prove it.
I'm fairly sure that it's equivalent to show that the topology on $X$ is the same as the product topology on $V \oplus W$ (since in a preadditive category products=coproducts=biproducts)
I'm interested in the result for an application to the space of test functions on a domain, which has a lot more structure than a general topological vector space (it's locally convex, locally bounded and has the Heine-Borel property but is not metrisable), so if anyone knows of/can think of a proof in that special case that would be great also.

REPLY [2 votes]: When $X$ is an $F-$space (in particular when $X$ is Banach) by the Closed Graph Theorem the claim follows. 
See Linear Operators Part I by Dunford and Schwartz VI.3 and Theorem II.2.4 for clarifications.