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Prove that a linear mapping from a normed space into a normed space is continuous if and only if it maps bounded sets to bounded sets.

I have an idea if the sets were Cauchy, but I can't assume that here. Can I still use it? I'm having some issues with this since the definition of a linear operator is a bounded operator. Any help/hints would be much appreciated.

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Let $A:U\rightarrow V$ be bounded. If $X \subset U$ is bounded there exists a $K \in \mathbb{R}$ such that $||x|| < K$ for all $x \in X$. Let $y \in A(X)$ i.e. $Ax = y$ for an $x \in X$ then

$||y|| = ||Ax|| \leq ||A||\cdot ||x|| \leq ||A||\cdot K$ therefore $A(X)$ is bounded too.

Now let $A$ send bounded sets to bounded sets and let $K$ be a bound for the image of the unit ball under $A$ then

$||A|| = \sup_{||x|| \leq 1} ||Ax|| \leq K$, i.e. $A$ is bounded.

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