# Prove that $T_1+T_2$ is a closed linear operator

Let $X$ and $Y$ be normed spaces. If $T_1$ :$X$ $\rightarrow$ $Y$ is a closed linear operator and $T_2$ $\in$ $BL(X,Y)$, show that $T_1+T_2$ is a closed linear operator.

I'm stuck with this problem. Thanks in advance!

Hint: Consider a convergent sequence $x_n\rightarrow x$ such that $(T_1+T_2)x_n\rightarrow y$ for some $y\in Y$. You wish to show that $x\in \mathcal D(T_1)$ and $T_1 x=y-T_2x$.
• Given that $T_1$ is a closed operator, what is a possible strategy to show that an object is in the domain of $T_1$? – Jonas Dahlbæk Sep 25 '14 at 10:20