Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be normed spaces over the same scalar field $\mathbb{F}$, where $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$. Let $\mathcal{L}(V,W)$ denote the set of all bounded $\mathbb{F}$-linear maps $T:V\to W$. Define pointwise operations on $\mathcal{L}(V,W)$ by
Then $\mathcal{L}(V,W)$ is a [vector space](/page/Vector%20Space) over $\mathbb{F}$ under these operations, and $\|\cdot\|_{\mathcal{L}(V,W)}$ is a norm on $\mathcal{L}(V,W)$.