Let $X$ and $Y$ be normed linear spaces and let $x(\neq 0)\in X,y(\neq 0)\in Y$. I want to show that there exists a bounded linear operator $T:X\to Y$ such that $Tx=y$.
Let for every $T\in B(X,Y)$, $Tx\neq y$. Then by Hahn-Banach theorem there exists $f\in X^*$ such that $f(Tx)\neq f(y)$. But here I am not getting any contradiction. Please suggest!