If $S$ is symmetric, then its image of linear operator $Λ(S)$ is symmetric.

Question

Let $X, Y$ be Banach spaces over the real numbers, and let $Λ : X → Y$ be a bounded linear operator. Prove that If $S$ is symmetric, then its image $Λ(S)$ is symmetric.

My attempt:

I have some problem to know where we need the boundedness of the operator and completeness of the spaces is needed?. Or my solution has flaws?

Let $-a,a \in S$ then $\Lambda a ,\Lambda(-a) \in \Lambda (S)$. We can also write $\Lambda (S) \ni \Lambda(-a) = -\Lambda a$ so it is symmetric.

Answer 1

This requires only the fact that $\Lambda$ is linear but the way you have written the proof is not correct. Pick any point $y$ in $\Lambda (S)$. Then $y=\Lambda (x)$ for some $x \in S$. By hyptothesis $-x \in S$ so $-y=-\Lambda (x)=\Lambda (-x) \in \Lambda (S)$.