F-space

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, an F-space is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 over the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers together with a <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> 
 
 
 
 d
 :
 X
 ×
 X
 →
 
 R
 
 
 
 {\displaystyle d:X\times X\to \mathbb {R} }
 
 such that

<ol><li>Scalar multiplication in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> with respect to 
 
 
 
 d
 
 
 {\displaystyle d}
 
 and the standard metric on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 or 
 
 
 
 
 C
 
 .
 
 
 {\displaystyle \mathbb {C} .}
 
</li>
<li>Addition in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is continuous with respect to 
 
 
 
 d
 .
 
 
 {\displaystyle d.}
 
</li>
<li>The metric is <a href="/facts/Translation-invariant_metric/RkUptSo8">translation-invariant</a>; that is, 
 
 
 
 d
 (
 x
 +
 a
 ,
 y
 +
 a
 )
 =
 d
 (
 x
 ,
 y
 )
 
 
 {\displaystyle d(x+a,y+a)=d(x,y)}
 
 for all 
 
 
 
 x
 ,
 y
 ,
 a
 ∈
 X
 .
 
 
 {\displaystyle x,y,a\in X.}
 
</li>
<li>The metric space 
 
 
 
 (
 X
 ,
 d
 )
 
 
 {\displaystyle (X,d)}
 
 is <a href="/facts/Complete_metric_space/lsckmU8G">complete</a>.</li></ol>
The operation 
 
 
 
 x
 ↦
 ‖
 x
 ‖
 :=
 d
 (
 0
 ,
 x
 )
 
 
 {\displaystyle x\mapsto \|x\|:=d(0,x)}
 
 is called an F-norm, although in general an F-norm is not required to be homogeneous. By <a href="/facts/Translation_invariance/PYVMaqWb">translation-invariance</a>, 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 use the term <a href="/facts/Fr%25C3%25A9chet_space/9gXkQu2u">Fréchet space</a> rather than F-space, but usually the term "Fréchet space" is reserved for <a href="/facts/Locally_convex_topological_vector_space/AGrPrayC">locally convex</a> F-spaces. 
Some other authors use the term "F-space" as a synonym of "Fréchet space", by which they mean a locally convex complete metrizable <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a>. 
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 <a href="/facts/Metrizable_topological_vector_space/6FUHfx6W">metrizable</a> in a manner that satisfies the above properties.

F-space open-in-new

F-space