Isomorphism between $E$ and $Lin(E)$ : infinite dimensional case.

Question

Does $E$ and $Lin(E)$ where $Lin(E)=\{A : E \rightarrow E ∣ A \quad\text{is linear}\}$ are isomorph if $E$ infinite dimensional case ?

I know that if $E$ is finite dimension the result is true if and only if the dimension is $0$ or $1$ . I am tempted to say the result is false because they are not equipotent by Kaplansky theorem.

Answer 1

No. Let $B$ be a Hamel basis of $E$. Each function $f\colon B\to E$ extends uniquely to a linear map. By Cantor's theorem $|E|^{|B|}>|B|=\dim E$.