Continuous linear map

Definition

Let $V$ and $W$ be vector spaces over $\mathbb{R}$ or $\mathbb{C}$, with norms $\|\cdot\|_V$ and $\|\cdot\|_W$. A map $T \colon V \to W$ is a continuous linear map when $T$ is linear and continuous for the metrics $\|x - y\|_V$ and $\|x - y\|_W$.

Remarks

A linear map is continuous if and only if it is bounded, that is, $\|Tx\|_W \le C \|x\|_V$ for some constant $C$ and all $x$. This is bounded linear maps are continuous. The smallest such $C$ is the operator norm of $T$. On a finite-dimensional space, each linear map is continuous. The derivative $p \mapsto p'$ on polynomials, with the sup norm on $[0, 1]$, is linear but not continuous: $\|x^n\|_\infty = 1$ but $\|n x^{n-1}\|_\infty = n$.

Explanations

L1 human

A continuous linear map respects the vector space structure and also the notion of closeness. In infinite dimensions, a linear map can fail to be continuous, so this is a real condition.