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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!

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Construct a bounded linear functional $T: X\to \mathbb R$ so that $T(x) = 1$ (Consequence of Hahn Banach Theorem). Then the bounded linear operator you want is $\iota \circ T$, where $\iota :\mathbb R \to Y$ is given by $\iota(s) = sy$.

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