FK-space

Remarks

Example: the space $\omega$ of all sequences, with the product topology of $\mathbb{K}^{\N}$, is an FK-space. This topology is given by the seminorms $p_n(x) = \abs{x_n}$, so it is locally convex and makes each coordinate continuous. It is metrizable by $d(x, y) = \sum_n 2^{-n} \min(1, \abs{x_n - y_n})$, and it is complete because a Cauchy sequence converges in each coordinate.

Example: each space ℓ^p with $1 \le p \le \infty$ and its usual norm is an FK-space, since $\abs{x_n} \le \norm{x}_p$.

Non-example: the space of sequences with finitely many nonzero terms, with the norm $\sup_n \abs{x_n}$, is not an FK-space under this norm, because it is not complete. The sequences $(1, 1/2, \dots, 1/k, 0, 0, \dots)$ form a Cauchy sequence whose coordinatewise limit $(1/n)$ is not in the space.

An FK-space whose topology is given by a complete norm is a BK-space. By the closed graph theorem, a sequence space has at most one FK topology, and every inclusion map between FK-spaces is continuous.