Let $X,Y$ be normed spaces and $T:X\to Y$ is an open linear map. Show that $T$ is surjective.
In order to show $T$ is surjective let's take $y_0\in Y$ and assume the contrary that $Tx\neq y_0\forall x\in X$.
Now taking $x_0\in X\implies Tx_0\neq y$.
Also $T(B(x_0,r))$ is open. $X=\cup_{n\in \Bbb N}B(x_0,n)\implies T(X)\subset \cup_{n\in \Bbb N} T(B(x_0,n))$.
I am unable to find any contradiction.Can someone kindly help?