Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be normed spaces over the same scalar field $\mathbb{F}$, where $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$. If $V$ is finite-dimensional over $\mathbb{F}$, then every [linear map](/page/Linear%20Map) $T:V\to W$ is bounded. Equivalently, for every linear map $T:V\to W$, there exists a constant $C\geq 0$ such that

\begin{align*} \|T v\|_W\leq C\|v\|_V \end{align*}

for every $v\in V$.