Self Study - NOT HOMEWORK OR GRADED IN ANYWAY
I just want to know if my proof is correct.
Statement :
Let $X$ and $Y$ be normed linear spaces and let $T:X \to Y$ and $S:X \to Y$ be completely continuous linear operators. Show that $T+S:X \to Y$ is linear and completely continuous.
The definition I am using is that $T:X \to Y$ is a completely continuous linear operator if T is linear and maps bounded sets into relatively compact sets.
Proof :
Let $M$ be a bounded subset of $X.$ If $y \in (T_1+T_2)(M),$ then there exists $x \in M$ such that $y=(T_1+T_2)(x)=T_1(x)+T_2(x).$
Thus, $y \in T_1(M)+T_2(M).$
We now show that $T_1(M)+T_2(M) \subset \overline{T_1(M)} +\overline{T_2(M)}.$
If $y \in T_1(M)+T_2(M)$ then we have that there exists $x \in M$ such that $y=T_1(x)+T_2(x).$
But then $T_1(x) \in \overline{T_1(M)}$ and $T_2(x) \in \overline{T_2(M)}$ so that $y \in \overline{T_1(M)} +\overline{T_2(M)}.$
We then have that $\overline{T_1(M)+T_2(M)}\subset \overline{T_1(M)} +\overline{T_2(M)}.$
Then
$$\overline{(T_1+T_2)(M)}\subset \overline{T_1(M)+T_2(M)} \subset \overline{T_1(M)} +\overline{T_2(M)}.$$
Then $\overline{(T_1+T_2)(M)}$ is a closed subset of a compact space, since $\overline{T_1(M)}$ and $\overline{T_2(M)}$ are compact and the sum of compact sets is compact.
We thus have that $\overline{(T_1+T_2)(M)}$ is compact.
Since the sum of linear operators is linear, we are done. $\blacksquare$