An Equation about Continuous Linear Operators $T \overline{\text{span}(A)} = \overline{T\text{span}(A)}$ in Normed Spaces

Question

Let $X$ and $Y$ be normed spaces, let $T \in B(X,Y)$ where $B(X,Y)$ represents all continuous linear operators from $X$ to $Y$. $\DeclareMathOperator{\span}{span}$ Let $A \subset X$ and $\span(A)$ is the smallest linear subspace which contains $A$. Prove that $T \overline{\span(A)} = \overline{T\span(A)}$.

Actually, I want the proof for $\overline{T\span(A)} \subset T \overline{\span(A)} $, since the opposite direction can be easily obtained by the continuity and linearity of $T$.

Answer 1

It's not true. The range of a continuous linear operator in infinite-dimensional normed spaces is not necessarily closed.