F-space
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In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, an F-space is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
X over the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers together with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
d : X \times X \to \R such that # Scalar multiplication in X is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
with respect to d and the standard metric on \R or \Complex. # Addition in X is continuous with respect to d. # The metric is translation-invariant; that is, d(x + a, y + a) = d(x, y) for all x, y, a \in X. # The metric space (X, d) is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. The operation x \mapsto \, x\, := d(0, x) 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 rather than , 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 In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. 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 topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
in a manner that satisfies the above properties.


Examples

All
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s and
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s are F-spaces. In particular, a Banach space is an F-space with an additional requirement that d(a x, 0) = , a, d(x, 0).Dunford N., Schwartz J.T. (1958). Linear operators. Part I: general theory. Interscience publishers, inc., New York. p. 59 The Lp spaces can be made into F-spaces for all p \geq 0 and for p \geq 1 they can be made into locally convex and thus Fréchet spaces and even Banach spaces.


Example 1

L^ ,\, 1/math> is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
.


Example 2

Let W_p(\mathbb) be the space of all complex valued
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
f(z) = \sum_ a_n z^n on the unit disc \mathbb such that \sum_n \left, a_n\^p < \infty then for 0 < p < 1, W_p(\mathbb) are F-spaces under the
p-norm In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki ...
: \, f\, _p = \sum_n \left, a_n\^p \qquad (0 < p < 1). In fact, W_p is a quasi-Banach algebra. Moreover, for any \zeta with , \zeta, \leq 1 the map f \mapsto f(\zeta) is a bounded linear (multiplicative functional) on W_p(\mathbb).


Sufficient conditions


Related properties

The open mapping theorem implies that if \tau \text \tau_2 are topologies on X that make both (X, \tau) and \left(X, \tau_2\right) into
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
metrizable topological vector space In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
s (for example, Banach or
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s) and if one topology is finer or coarser than the other then they must be equal (that is, if \tau \subseteq \tau_2 \text \tau_2 \subseteq \tau \text \tau = \tau_2).


See also

* * * * * * * * * * * * * *


References


Notes


Sources

* * * * * * * * {{TopologicalVectorSpaces pl:Przestrzeń Frécheta (analiza funkcjonalna)