The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following lemma
Let $T:\mathcal{D}(T)\to Y$ be a bounded linear operator with domain $\mathcal{D}(T)\subset X$, where $X$ and $Y$ are normed spaces. Then:
(a) If $\mathcal{D}(T)$ is a closed subset of $X$, then $T$ is closed.
(b) If $T$ is closed and $Y$ is complete, then $\mathcal{D}(T)$ is a closed subset of $X$. (page 295)
and the following problem
Let $X$ and $Y$ be normed spaces. If $T_1:X\to Y$ is a closed linear operator and $T_2\in\mathcal{B}(X,Y)$, show that $T_1 + T_2$ is a closed linear operator. (page 296)
If $X$ and $Y$ are Banach spaces, then we can conclude (by Closed Graph Theorem) that $T_1$ is bounded. So, $(T_1+T_2):X\to Y$ is bounded and thus it's closed (by lemma above).
My question is: how to solve it when $X$ and $Y$ are not necessarily complete spaces?
Thanks.