Definition:F-Space

Definition

Let $K$ be a topological field.

Let $\struct {X, \tau}$ be a topological vector space over $K$.

We say that $\struct {X, \tau}$ is an $F$-space if and only if there exits a metric $d$ on $X$ such that:

$(1): \quad$ $\tau$ is induced by $d$

$(2): \quad$ $d$ is an invariant metric

$(3): \quad$ $\struct {X, d}$ is a complete metric space.

With $d$ as above, we may also say $\struct {X, d}$ is an $F$-space.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.8$: Types of topological vector spaces