Let $X$ be a normed space, and $Y,Z$ subspaces of $X$. Then
$Y$ is algebraically complemented if the map $T:Y\times Z \to X$ given by $T(y,z)=y+z$ is bijective onto $X$ and moreover $Y$ is topologically complemented if this map $T$ is an isomorphism
Then I know above condition is equivalent to
Let $Y$ be subspace of $X$. Then $Y$ is topologically complemented iff there is a bounded projection $\pi$ from $X$ onto $Y$
Then it is stated that from this it is clear that any topologically complemented subspace must be closed. But I don't quite follow that - all this seems to tell me is that $Y$ is an image of bounded projection. I think I'm missing some obvious points. Can someone help me understand this?
Note that $1-\pi: X\to X$ is continuous (since it is bounded) and $$(1-\pi)(x)=0\iff \pi(x)=x \iff x\in Y.$$ So $(1-\pi)^{-1}(\{0\})=Y$ is closed as the preimage of a closed set.