Fréchet combination

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 locally convex metrizable TVS.

Pseudometrics and metrics

A pseudometric on a set $X$ is a map $d : X \times X \rarr \R$ satisfying the following properties:

1. $d(x, x) = 0 \text x \in X$;


2. Symmetry: $d(x, y) = d(y, x) \text x, y \in X$;


3. Subadditivity: $d(x, z) \leq d(x, y) + d(y, z) \text x, y, z \in X.$

A pseudometric is called a metric if it satisfies:

4. Identity of indiscernibles: for all $x, y \in X,$ if $d(x, y) = 0$ then $x = y.$

Ultrapseudometric

A pseudometric $d$ on $X$ is called a ultrapseudometric or a strong pseudometric if it satisfies:

5. Strong/Ultrametric triangle inequality: $d(x, z) \leq \max \ \text x, y, z \in X.$

Pseudometric space

A pseudometric space is a pair $(X, d)$ consisting of a set $X$ and a pseudometric $d$ on $X$ such that $X$'s topology is identical to the topology on $X$ induced by $d.$ We call a pseudometric space $(X, d)$ a metric space (resp. ultrapseudometric space) when $d$ is a metric (resp. ultrapseudometric).

Topology induced by a pseudometric

If $d$ is a pseudometric on a set $X$ then collection of open balls:
$B_r(z) := \$ as $z$ ranges over $X$ and $r > 0$ ranges over the positive real numbers,
forms a basis for a topology on $X$ that is called the $d$-topology or the pseudometric topology on $X$ induced by $d.$

:: If $(X, d)$ is a pseudometric space and $X$ is treated as a topological space, then unless indicated otherwise, it should be assumed that $X$ is endowed with the topology induced by $d.$

Pseudometrizable space

A topological space $(X, \tau)$ is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) $d$ on $X$ such that $\tau$ is equal to the topology induced by $d.$

Pseudometrics and values on topological groups

An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.

A topology $\tau$ on a real or complex vector space $X$ is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes $X$ into a topological vector space).

Every topological vector space (TVS) $X$ is an additive commutative topological group but not all group topologies on $X$ are vector topologies.
This is because despite it making addition and negation continuous, a group topology on a vector space $X$ may fail to make scalar multiplication continuous.
For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

Translation invariant pseudometrics

If $X$ is an additive group then we say that a pseudometric $d$ on $X$ is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

1. Translation invariance: $d(x + z, y + z) = d(x, y) \text x, y, z \in X$;


2. $d(x, y) = d(x - y, 0) \text x, y \in X.$

Value/G-seminorm

If $X$ is a topological group the a value or G-seminorm on $X$ (the ''G'' stands for Group) is a real-valued map $p : X \rarr \R$ with the following properties:

1. Non-negative: $p \geq 0.$


2. Subadditive: $p(x + y) \leq p(x) + p(y) \text x, y \in X$;


3. $p(0) = 0..$


4. Symmetric: $p(-x) = p(x) \text x \in X.$

where we call a G-seminorm a G-norm if it satisfies the additional condition:

5. Total/Positive definite: If $p(x) = 0$ then $x = 0.$

Properties of values

If $p$ is a value on a vector space $X$ then:

• $, p(x) - p(y), \leq p(x - y) \text x, y \in X.$


• $p(n x) \leq n p(x)$ and $\frac p(x) \leq p(x / n)$ for all $x \in X$ and positive integers $n.$


• The set $\$ is an additive subgroup of $X.$

Equivalence on topological groups

Pseudometrizable topological groups

An invariant pseudometric that doesn't induce a vector topology

Let $X$ be a non-trivial (i.e. $X \neq \$) real or complex vector space and let $d$ be the translation-invariant trivial metric on $X$ defined by $d(x, x) = 0$ and $d(x, y) = 1 \text x, y \in X$ such that $x \neq y.$
The topology $\tau$ that $d$ induces on $X$ is the discrete topology, which makes $(X, \tau)$ into a commutative topological group under addition but does form a vector topology on $X$ because $(X, \tau)$ is disconnected but every vector topology is connected.
What fails is that scalar multiplication isn't continuous on $(X, \tau).$

This example shows that a translation-invariant (pseudo)metric is enough to guarantee a vector topology, which leads us to define paranorms and ''F''-seminorms.

Additive sequences

A collection $\mathcal$ of subsets of a vector space is called additive if for every $N \in \mathcal,$ there exists some $U \in \mathcal$ such that $U + U \subseteq N.$

All of the above conditions are consequently a necessary for a topology to form a vector topology.
Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions.
These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Assume that $n_ = \left(n_1, \ldots, n_k\right)$ always denotes a finite sequence of non-negative integers and use the notation:
$\sum 2^ := 2^ + \cdots + 2^ \quad \text \quad \sum U_ := U_ + \cdots + U_.$

For any integers $n \geq 0$ and $d > 2,$
$U_n \supseteq U_ + U_ \supseteq U_ + U_ + U_ \supseteq U_ + U_ + \cdots + U_ + U_ + U_.$

From this it follows that if $n_ = \left(n_1, \ldots, n_k\right)$ consists of distinct positive integers then $\sum U_ \subseteq U_.$

It will now be shown by induction on $k$ that if $n_ = \left(n_1, \ldots, n_k\right)$ consists of non-negative integers such that $\sum 2^ \leq 2^$ for some integer $M \geq 0$ then $\sum U_ \subseteq U_M.$
This is clearly true for $k = 1$ and $k = 2$ so assume that $k > 2,$ which implies that all $n_i$ are positive.
If all $n_i$ are distinct then this step is done, and otherwise pick distinct indices $i < j$ such that $n_i = n_j$ and construct $m_ = \left(m_1, \ldots, m_\right)$ from $n_$ by replacing each $n_i$ with $n_i - 1$ and deleting the $j^$ element of $n_$ (all other elements of $n_$ are transferred to $m_$ unchanged).
Observe that $\sum 2^ = \sum 2^$ and $\sum U_ \subseteq \sum U_$ (because $U_ + U_ \subseteq U_$) so by appealing to the inductive hypothesis we conclude that $\sum U_ \subseteq \sum U_ \subseteq U_M,$ as desired.

It is clear that $f(0) = 0$ and that $0 \leq f \leq 1$ so to prove that $f$ is subadditive, it suffices to prove that $f(x + y) \leq f(x) + f(y)$ when $x, y \in X$ are such that $f(x) + f(y) < 1,$ which implies that $x, y \in U_0.$
This is an exercise.
If all $U_i$ are symmetric then $x \in \sum U_$ if and only if $- x \in \sum U_$ from which it follows that $f(-x) \leq f(x)$ and $f(-x) \geq f(x).$
If all $U_i$ are balanced then the inequality $f(s x) \leq f(x)$ for all unit scalars $s$ such that $, s, \leq 1$ is proved similarly.
Because $f$ is a nonnegative subadditive function satisfying $f(0) = 0,$ as described in the article on sublinear functionals, $f$ is uniformly continuous on $X$ if and only if $f$ is continuous at the origin.
If all $U_i$ are neighborhoods of the origin then for any real $r > 0,$ pick an integer $M > 1$ such that $2^ < r$ so that $x \in U_M$ implies $f(x) \leq 2^ < r.$
If the set of all $U_i$ form basis of balanced neighborhoods of the origin then it may be shown that for any $n > 1,$ there exists some $0 < r \leq 2^$ such that $f(x) < r$ implies $x \in U_n.$ $\blacksquare$

Paranorms

If $X$ is a vector space over the real or complex numbers then a paranorm on $X$ is a G-seminorm (defined above) $p : X \rarr \R$ on $X$ that satisfies any of the following additional conditions, each of which begins with "for all sequences $x_ = \left(x_i\right)_^$ in $X$ and all convergent sequences of scalars $s_ = \left(s_i\right)_^$":

1. Continuity of multiplication: if $s$ is a scalar and $x \in X$ are such that $p\left(x_i - x\right) \to 0$ and $s_ \to s,$ then $p\left(s_i x_i - s x\right) \to 0.$


2. Both of the conditions:
   * if $s_ \to 0$ and if $x \in X$ is such that $p\left(x_i - x\right) \to 0$ then $p\left(s_i x_i\right) \to 0$;
   * if $p\left(x_\right) \to 0$ then $p\left(s x_i\right) \to 0$ for every scalar $s.$


3. Both of the conditions:
   * if $p\left(x_\right) \to 0$ and $s_ \to s$ for some scalar $s$ then $p\left(s_i x_i\right) \to 0$;
   * if $s_ \to 0$ then $p\left(s_i x\right) \to 0 \text x \in X.$


4. Separate continuity:
   * if $s_ \to s$ for some scalar $s$ then $p\left(s x_i - s x\right) \to 0$ for every $x \in X$;
   * if $s$ is a scalar, $x \in X,$ and $p\left(x_i - x\right) \to 0$ then $p\left(s x_i - s x\right) \to 0$ .

A paranorm is called total if in addition it satisfies:

• Total/Positive definite: $p(x) = 0$ implies $x = 0.$

Properties of paranorms

If $p$ is a paranorm on a vector space $X$ then the map $d : X \times X \rarr \R$ defined by $d(x, y) := p(x - y)$ is a translation-invariant pseudometric on $X$ that defines a on $X.$

If $p$ is a paranorm on a vector space $X$ then:

• the set $\$ is a vector subspace of $X.$


• $p(x + n) = p(x) \text x, n \in X$ with $p(n) = 0.$


• If a paranorm $p$ satisfies $p(s x) \leq , s, p(x) \text x \in X$ and scalars $s,$ then $p$ is absolutely homogeneity (i.e. equality holds) and thus $p$ is a seminorm.

Examples of paranorms

• If $d$ is a translation-invariant pseudometric on a vector space $X$ that induces a vector topology $\tau$ on $X$ (i.e. $(X, \tau)$ is a TVS) then the map $p(x) := d(x - y, 0)$ defines a continuous paranorm on $(X, \tau)$; moreover, the topology that this paranorm $p$ defines in $X$ is $\tau.$


• If $p$ is a paranorm on $X$ then so is the map q(x) := p(x) / + p(x)


• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).


• Every seminorm is a paranorm.


• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).


• The sum of two paranorms is a paranorm.


• If $p$ and $q$ are paranorms on $X$ then so is $(p \wedge q)(x) := \inf_ \.$ Moreover, $(p \wedge q) \leq p$ and $(p \wedge q) \leq q.$ This makes the set of paranorms on $X$ into a conditionally complete lattice.


• Each of the following real-valued maps are paranorms on $X := \R^2$:
  * $(x, y) \mapsto , x,$
  * $(x, y) \mapsto , x, + , y,$


• The real-valued maps $(x, y) \mapsto \sqrt$ and $(x, y) \mapsto \left, x^2 - y^2\^$ are a paranorms on $X := \R^2.$


• If $x_ = \left(x_i\right)_$ is a Hamel basis on a vector space $X$ then the real-valued map that sends $x = \sum_ s_i x_i \in X$ (where all but finitely many of the scalars $s_i$ are 0) to $\sum_ \sqrt$ is a paranorm on $X,$ which satisfies $p(sx) = \sqrt p(x)$ for all $x \in X$ and scalars $s.$


• The function $p(x) := , \sin (\pi x), + \min \$is a paranorm on $\R$ that is balanced but nevertheless equivalent to the usual norm on $R.$ Note that the function $x \mapsto , \sin (\pi x),$ is subadditive.


• Let $X_$ be a complex vector space and let $X_$ denote $X_$ considered as a vector space over $\R.$ Any paranorm on $X_$ is also a paranorm on $X_.$


• 
  

''F''-seminorms

If $X$ is a vector space over the real or complex numbers then an ''F''-seminorm on $X$ (the $F$ stands for Fréchet) is a real-valued map $p : X \rarr \R$ with the following properties:

1. Non-negative: $p \geq 0.$


2. Subadditive: $p(x + y) \leq p(x) + p(y) \text x, y \in X$;


3. Balanced: $p(a x) \leq p(x)$ for all $x \in X$ and all scalars $a$ satisfying $, a, \leq 1$;
   * This condition guarantees that each set of the form $\$ or $\$ for some $r \geq 0$ is balanced.


4. for every $x \in X,$ $p\left(\frac x\right) \to 0$ as $n \to \infty$
   * The sequence $\left(\frac\right)_^$ can be replaced by any positive sequence converging to 0.

An ''F''-seminorm is called an ''F''-norm if in addition it satisfies:

5. Total/Positive definite: $p(x) = 0$ implies $x = 0.$

An ''F''-seminorm is called monotone if it satisfies:

6. Monotone: $p(r x) < p(s x)$ for all non-zero $x \in X$ and all real $s$ and $t$ such that $s < t.$

''F''-seminormed spaces

An ''F''-seminormed space (resp. ''F''-normed space) is a pair $(X, p)$ consisting of a vector space $X$ and an ''F''-seminorm (resp. ''F''-norm) $p$ on $X.$

If $(X, p)$ and $(Z, q)$ are ''F''-seminormed spaces then a map $f : X \to Z$ is called an isometric embedding if $q(f(x) - f(y)) = p(x, y) \text x, y \in X.$

Every isometric embedding of one ''F''-seminormed space into another is a topological embedding, but the converse is not true in general.

Examples of ''F''-seminorms

• Every positive scalar multiple of an ''F''-seminorm (resp. ''F''-norm, seminorm) is again an ''F''-seminorm (resp. ''F''-norm, seminorm).


• The sum of finitely many ''F''-seminorms (resp. ''F''-norms) is an ''F''-seminorm (resp. ''F''-norm).


• If $p$ and $q$ are ''F''-seminorms on $X$ then so is their pointwise supremum $x \mapsto \sup \.$ The same is true of the supremum of any non-empty finite family of ''F''-seminorms on $X.$


• The restriction of an ''F''-seminorm (resp. ''F''-norm) to a vector subspace is an ''F''-seminorm (resp. ''F''-norm).


• A non-negative real-valued function on $X$ is a seminorm if and only if it is a convex ''F''-seminorm, or equivalently, if and only if it is a convex balanced ''G''-seminorm. In particular, every seminorm is an ''F''-seminorm.


• For any $0 < p < 1,$ the map $f$ on $\R^n$ defined by
  \left(x_1, \ldots, x_n\right)p = \left, x_1\^p + \cdots \left, x_n\^p
  is an ''F''-norm that is not a norm.


• If $L : X \to Y$ is a linear map and if $q$ is an ''F''-seminorm on $Y,$ then $q \circ L$ is an ''F''-seminorm on $X.$


• Let $X_$ be a complex vector space and let $X_$ denote $X_$ considered as a vector space over $\R.$ Any ''F''-seminorm on $X_$ is also an ''F''-seminorm on $X_.$

Properties of ''F''-seminorms

Every ''F''-seminorm is a paranorm and every paranorm is equivalent to some ''F''-seminorm.
Every ''F''-seminorm on a vector space $X$ is a value on $X.$ In particular, $p(x) = 0,$ and $p(x) = p(-x)$ for all $x \in X.$

Topology induced by a single ''F''-seminorm

Topology induced by a family of ''F''-seminorms

Suppose that $\mathcal$ is a non-empty collection of ''F''-seminorms on a vector space $X$ and for any finite subset $\mathcal \subseteq \mathcal$ and any $r > 0,$ let
$U_ := \bigcap_ \.$

The set $\left\$ forms a filter base on $X$ that also forms a neighborhood basis at the origin for a vector topology on $X$ denoted by $\tau_.$ Each $U_$ is a balanced and absorbing subset of $X.$ These sets satisfy
$U_ + U_ \subseteq U_.$

• $\tau_$ is the coarsest vector topology on $X$ making each $p \in \mathcal$ continuous.


• $\tau_$ is Hausdorff if and only if for every non-zero $x \in X,$ there exists some $p \in \mathcal$ such that $p(x) > 0.$


• If $\mathcal$ is the set of all continuous ''F''-seminorms on $\left(X, \tau_\right)$ then $\tau_ = \tau_.$


• If $\mathcal$ is the set of all pointwise suprema of non-empty finite subsets of $\mathcal$ of $\mathcal$ then $\mathcal$ is a directed family of ''F''-seminorms and $\tau_ = \tau_.$

Fréchet combination

Suppose that $p_ = \left(p_i\right)_^$ is a family of non-negative subadditive functions on a vector space $X.$

The Fréchet combination of $p_$ is defined to be the real-valued map
$p(x) := \sum_^ \frac.$

As an ''F''-seminorm

Assume that $p_ = \left(p_i\right)_^$ is an increasing sequence of seminorms on $X$ and let $p$ be the Fréchet combination of $p_.$
Then $p$ is an ''F''-seminorm on $X$ that induces the same locally convex topology as the family $p_$ of seminorms.

Since $p_ = \left(p_i\right)_^$ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form $\left\$ as $i$ ranges over all positive integers and $r > 0$ ranges over all positive real numbers.

The translation invariant pseudometric on $X$ induced by this ''F''-seminorm $p$ is
$d(x, y) = \sum^_ \frac \frac.$

This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.

As a paranorm

If each $p_i$ is a paranorm then so is $p$ and moreover, $p$ induces the same topology on $X$ as the family $p_$ of paranorms.
This is also true of the following paranorms on $X$:

• $q(x) := \inf_ \left\.$


• $r(x) := \sum_^ \min \left\.$

Generalization

The Fréchet combination can be generalized by use of a bounded remetrization function.

A bounded remetrization function is a continuous non-negative non-decreasing map R : subadditive_In_mathematics,_subadditivity_is_a_property_of_a_function_that_states,_roughly,_that_evaluating_the_function_for_the_sum_of_two__elements_of_the_domain_always_returns_something_less_than_or_equal_to_the_sum_of_the_function's_values_at_each_element.__...