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

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

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