in this case i'm not assuming that $\displaystyle T$ is linear (this is because i use the word "function"). It's true that in general vector spaces the first statement doesn't implies the second (i agree with you that it would be indicated on the definition), but i'm thinking in the special case when $\displaystyle T:\mathbb{R}^n\to\mathbb{R}^m$.

Fortunately, i found a negative answer to that implication but the construction of such a function requires the axiom of choice, so we don't have an explicit function.