I'm reading through some functional analysis lecture notes and there the closed graph theorem was stated in the following form:
Let $X$ be a Baire locally convex space and $Y$ a Frechet space. If the graph of a linear map between $X$ and $Y$ is closed in $X\times Y$, then this map is continuous.
The text goes than on to say that if just one of these spaces in Banach, the theorem doesn't hold anymore. But how can this be true ? Banach spaces are in particular locally convex and Frechet, so the theorem would have to hold.
If someone indeed somehow comes up with what I'm not seeing here could he/she please provide the construction of this counterexample ?