Bounded linear maps are continuous

Statement

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be linear. Then $T$ is continuous if and only if $T$ is bounded, that is, there is a real $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$.

Remarks

If $T$ is bounded, then $\|Tx - Ty\|_W = \|T(x - y)\|_W \le C \|x - y\|_V$. Thus $T$ is Lipschitz, and so uniformly continuous. Conversely, if $T$ is continuous at $0$, there is $\delta > 0$ with $\|Tx\|_W \le 1$ when $\|x\|_V \le \delta$. By linearity, $\|Tx\|_W \le \|x\|_V / \delta$ for all $x$. Thus, for a linear map, continuity at one point already gives continuity everywhere. The name bounded means that $T$ maps bounded sets to bounded sets. It does not mean that $T(V)$ is bounded.

Explanations

L1 human

For linear maps, continuity is the same as a simple estimate. The length of $Tx$ is at most a fixed multiple of the length of $x$.