BK-space

Remarks

Equivalently, a BK-space is an FK-space whose topology is given by a norm.

Example: $\ell^\infty$ with $\norm{x}_\infty = \sup_n \abs{x_n}$ is a BK-space. It is a Banach space, and $\abs{x_n} \le \norm{x}_\infty$ shows that each coordinate map is continuous. In the same way, the spaces ℓ^p for $1 \le p < \infty$, the space $c$ of convergent sequences and the space $c_0$ of null sequences are BK-spaces.

Non-example: the space $\omega$ of all sequences is an FK-space but not a BK-space. Its FK topology, the topology of coordinatewise convergence, is not given by any norm, because no neighbourhood of $0$ in it is bounded.