T(X) Fin. Co-D. Crit. for the Ran. of a BLO to be Cl., X and Y are Ban. Sp.

T(X) Finite Co-Dimensional Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces

Recall that if $X$ is a linear space and $M \subset X$ is a subspace then $M$ is said to be finite co-dimensional if $M$ has a finite-dimensional algebraic complement $M'$.

Also recall from the Topological Complement Criterion for the Range of a BLO to be Closed when X and Y are Banach Spaces page that if $X$ and $Y$ are Banach spaces and $T : X \to Y$ is a bounded linear operator then if $T(X)$ has a topological complement then $T(X)$ is closed.

We will now prove a very nice consequence of the above theorem.

Theorem 1: Let $X$ and $Y$ be Banach spaces and let $T : X \to Y$ be a bounded linear operator. If $T(X)$ is finite co-dimensional then $T(X)$ is closed.

• Proof: Let $T(X)$ be finite co-dimensional. Let $M \subset Y$ be a finite-dimensional algebraic complement of $T(X)$.

• Since $M \subseteq Y$ is finite-dimensional proper subspace of $Y$, we have that $M$ is closed. Therefore $T(X)$ has a topological complement.

• By the theorem referenced above, $T(X)$ must be closed. $\blacksquare$