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## Main Question or Discussion Point

Let X be a norm space, and X=Y+Z so that [itex]Y\cap Z=\{0\}[/itex]. Let P:X->Z be the projection [itex]y+z\mapsto z[/itex], when [itex]y\in Y[/itex] and [itex]z\in Z[/itex].

I see, that if P is continuous, then Y must be closed, because [itex]Y=P^{-1}(\{0\})[/itex].

Is the converse true? If Y is closed, does it make the projection continuous?

If yes, fine. If not, would finite dimensionality of Z make P continuous then?

I see, that if P is continuous, then Y must be closed, because [itex]Y=P^{-1}(\{0\})[/itex].

Is the converse true? If Y is closed, does it make the projection continuous?

If yes, fine. If not, would finite dimensionality of Z make P continuous then?