$X,Y$ infinite dimensional NLS , not both Banach , then $\exists T \in \mathcal L(X,Y)$ such that $R(T)$ is not closed in $Y$? Let $X,Y$ be infinite dimensional normed-linear spaces , not both Banach , then does there necessarily exist a continuous linear transformation $T:X \to Y $ such that $range (T)$ is not closed in $Y$ ? 

Let $X$ be a Banach space and $Y$ the increasing union of a sequence of finite-dimensional spaces $F_n$.  Then $X = \bigcup T^{-1}(F_n)$ with each $T^{-1}(F_n)$ closed.  By the Baire category theorem, some $T^{-1}(F_n)$ has
nonempty interior, and therefore it is all of $X$.  That is, $\text{Range}(T)$ is a subspace of $F_n$, and therefore is closed.
