# Sum of closed and bounded linear operators

Let $$T_1: X \rightarrow Y$$ be a closed linear operator and $$T_2: X \rightarrow Y$$ a bounded linear operator and $$X$$ and $$Y$$ normed spaces over the same field.

Is the sum of such operators also closed?

Here is my reasoning so far:

If we take a convergent sequence in the graph $$G(T_1 + T_2)$$, assuming such sequence exists, say $$z_n = (x_n, (T_1+T_2)x_n)$$ then it has to converge in their own pair of spaces, meaning, $$x_n \rightarrow x \in X$$ and $$(T_1 +T_2)x_n \rightarrow (T_1 +T_2)x \in R(T_1+T_2) \subset Y$$

Now, it seems that boundedness of $$T_2$$ is what guarantees that the the graph $$G$$ above will be closed. (Obviously along with linearity).

In this context, boundedness implies continuity and if there exists such $$x_n$$ then it has to converge to a point in the range of $$(T_1+T_2)$$.

I am having trouble understanding how I could use this fact though, namely $$\|T_2x\| \leq k\|x\|$$.

I would really appreciate hints on how to do so. Thanks in advance!

Take $$T_1$$ closed, $$T_2$$ bounded and $$x_n\to x \in X$$ so that $$(T_1+T_2)x_n \to y\in Y.$$ Then $$T_1x_n\to y-T_2x$$. (That step is where you use $$T_2$$ being continuous). $$T_1$$ is closed so $$T_1x=y-T_2x$$ and rearranged you have your result