If $X$ is an infinite dimensional hilbert space and $X_n$ be its $n$ dimensional subspace. Let $P$ denotes the orthogonal projection on $X_n$ from $X$. Also $T$ is a bounded linear operator from $X$ to hilbert space $Y$.
Now How to prove that range of this map $TP:X \to Y$ is closed.
I know the following results
For an orthogonal projection range$(P)$ is closed.
the range of $T$ is closed in $Y$ if $T$ is open.
But how to prove this i don't know.