A question in functional analysis about bounded linear operator.

Question

Suppose $Banach$ Space $E$ is the direct sum of its closed subspaces $L$、$M$, and $M$ is finite-dimensional, $T$ is a bounded linear operator from $E$ to itself. Prove that $T(E)$ is a closed subspace of $E$ iff $T(L)$ is closed. I don't know how to use the condition: $M$ is finite-dimensional.

Answer 1

A finite dimensional subspace of a Banach space is always closed with respect to the norm topology. It isn't necessarily true in the infinite dimensional case, so you are using the fact of the spaces finite dimension to conclude the space is closed.