Continuous linear transformation on normed linear spaces with $T(x)=y$

Question

Let $X,Y$ be normed linear spaces and let $x(\neq0)\in X,y(\neq0)\in Y$. prove that there exists $T\in \mathscr{B}(X,Y)$ such that $Tx=y$. I thought of a constant map but that will be not linear and continuous. Please give me idea how to construct this type of function? and thanks in advance

Answer 1

Hint:

Define a bounded linear functional $f : \operatorname{span}\{x\} \to \mathbb{K}$ with $f(\alpha x) = \alpha, \forall \alpha \in \mathbb{K}$.

Extend $f$ to a bounded linear functional $F : X \to \mathbb{K}$ by Hahn-Banach.

Check that the desired map $T : X \to Y$ is given by $Tz = F(z)y$.