Let $X,Y$ be a Banach space and let $T:X\rightarrow Y$ be a surjective continuous linear operator. As mentinoed in this stackexchange answer, we know there exists a right inverse linear operator.
But I want to know whether $T$ has a continuous right linear inverse $S$, i.e., there exists a bounded linear operator $S:Y\rightarrow X$ such that $T\circ S=\mathrm{id}_Y$.
In the case of finite dimensional vector spaces, linear operator is bounded linear operator. So there is nothing to prove.
This question is arisen when I study sum space of two Banach spaces.
Thanks in advance.
