Example

Let I be an arbitrary set of indices; it is known (GT, X, § 3.2) that a norm $\|x\|$ can be defined, on the set of bounded mappings $x = (\xi_i)$ of I in K, $\mathscr{B}(I; K)$ (also written $\mathscr{B}_K(I)$ or $\ell_K^\infty(I)$), by $\|x\| = \sup_{i \in I} |\xi_i|$. When I is a topological space, the set of bounded, continuous mappings of I in K is a closed subspace of the space $\mathscr{B}(I; K)$ (GT, X, § 3.1, cor. 2). Another subspace of $\mathscr{B}(I; K)$ is the set $\ell_K^1(I)$ of absolutely summable families $x = (\xi_i)$ (GT, X, § 3.6); we can define on this subspace another norm $\|x\|_1 = \sum_{i \in I} |\xi_i|$, that in general is not equivalent to the norm $\|x\| = \sup_{i \in I} |\xi_i|$ (I, p. 23, exerc. 6); when considering $\ell_K^1(I)$ as a normed space, without specifying its norm, it is always the norm $\|x\|$, that is meant. We write $\mathscr{B}(I)$ and $\ell^1(I)$ in place of $\mathscr{B}(I; \mathbf{R})$ and $\ell_R^1(I)$.