Topological vector space with a complete translation-invariant metric
In functional analysis, an F-space is a vector space
over the real or complex numbers together with a metric
such that
- Scalar multiplication in
is continuous with respect to
and the standard metric on
or ![{\displaystyle \mathbb {C} .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Addition in
is continuous with respect to ![{\displaystyle d.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- The metric is translation-invariant; that is,
for all ![{\displaystyle x,y,a\in X.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- The metric space
is complete.
The operation
is called an F-norm, although in general an F-norm is not required to be homogeneous. 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 use the term Fréchet space rather than F-space, but usually the term "Fréchet space" is reserved for locally convex 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 topological vector space.
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
[1]
The Lp spaces can be made into F-spaces for all
and for
they can be made into 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
are F-spaces under the p-norm:![{\displaystyle \|f\|_{p}=\sum _{n}\left|a_{n}\right|^{p}\qquad (0<p<1).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
In fact,
is a quasi-Banach algebra. Moreover, for any
with
the map
is a bounded linear (multiplicative functional) on ![{\displaystyle W_{p}(\mathbb {D} ).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Sufficient conditions
Related properties
The open mapping theorem implies that if
are topologies on
that make both
and
into complete metrizable topological vector spaces (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if
).
- A linear almost continuous map into an F-space whose graph is closed is continuous.
- A linear almost open map into an F-space whose graph is closed is necessarily an open map.
- A linear continuous almost open map from an F-space is necessarily an open map.
- A linear continuous almost open map from an F-space whose image is of the second category in the codomain is necessarily a surjective open map.
See also
References
- ^ Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59
- ^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
Notes
- ^ Not assume to be translation-invariant.
Sources
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