# F-space

In functional analysis, an **F-space** is a vector space *V* over the real or complex numbers together with a metric *d* : *V* × *V* → **R** so that

- Scalar multiplication in
*V*is continuous with respect to*d*and the standard metric on**R**or**C**. - Addition in
*V*is continuous with respect to*d*. - The metric is translation-invariant; i.e.,
*d*(*x*+*a*,*y*+*a*) =*d*(*x*,*y*) for all*x*,*y*and*a*in*V* - The metric space (
*V*,*d*) is complete.

The operation *x* ↦ ||*x*|| := d(0,*x*) is called an **F-norm**, although in general an F-norm is not required to be complete. By translation-invariance, the metric is recoverable from the F-norm. Thus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm.

Some authors call these spaces *Fréchet spaces*, but usually the term is reserved for locally convex F-spaces. The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be metrizable in a manner that satisfies the above properties.

## Examples

All Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that *d*(*αx*, 0) = |α|⋅*d*(*x*, 0).[1]

The L^{p} spaces are F-spaces for all p ≥ 0 and for p ≥ 1 they are locally convex and thus Fréchet spaces and even Banach spaces.

### Example 1

is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

### Example 2

Let be the space of all complex valued Taylor series

on the unit disc such that

then (for 0 < p < 1) are F-spaces under the p-norm:

In fact, is a quasi-Banach algebra. Moreover, for any with the map is a bounded linear (multiplicative functional) on .

## References

- Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59

- Rudin, Walter (1966),
*Real & Complex Analysis*, McGraw-Hill, ISBN 0-07-054234-1