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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

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

Some authors call these spaces "Fréchet spaces", but usually the term Fréchet space 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

Clearly, all Banach spaces and Fréchet spaces are F-spaces.

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. So for example, L^1/2[0,1] is a F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

References

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