Continuity of additive maps

Question

I studied all of previous posts about "when an additive map is continuous?" but I did not get my answer!

My question is the following:

Let $f:A\longrightarrow B$, be an bijective map from a Banach algebra $A$ onto a Banach algebra $B$ which has the following properties $f(0)=0$ and $f(I)=I$. If $f$ is additive (i.e. $f(X+Y)=f(X)+f(Y) \ \ \ \forall X,Y\in A$), then can we say $f$ is continuous at $0$?

If it is not true I am searching for a counterexample!

Thanks in advanced.

Answer 1

No. Consider $\mathbf{R}$ as a $\mathbf{Q}$- vector space. Consider a basis $\{e_i\}$ for $\mathbf{R}$, with $e_{i_0} = 1$. Let $f \colon \mathbf{R} \to \mathbf{R}$ be any $\mathbf{Q}$-linear map that permutes the basis elements in some way but preserves $e_{i_0}$. Then $f$ is a counterexample.

To see this, prove that the identity is the only additive map from $\mathbf{R}$ to itself which is continuous at $0$ and sends $1$ to $1$.