Metrizable topological vector space explained

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 x X\rarr\R

satisfying the following properties:
  1. d(x,x)=0forallx\inX

    ;
  2. Symmetry:

    d(x,y)=d(y,x)forallx,y\inX

    ;
  3. Subadditivity:

    d(x,z)\leqd(x,y)+d(y,z)forallx,y,z\inX.

A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all

    x,y\inX,

    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:
  1. Strong/Ultrametric triangle inequality:

    d(x,z)\leqmax\{d(x,y),d(y,z)\}forallx,y,z\inX.

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)forallx,y,z\inX

    ;
  2. d(x,y)=d(x-y,0)forallx,y\inX.

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\geq0.

  2. Subadditive:

    p(x+y)\leqp(x)+p(y)forallx,y\inX

    ;
  3. p(0)=0..

  4. Symmetric:

    p(-x)=p(x)forallx\inX.

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

  1. Total/Positive definite: If

    p(x)=0

    then

    x=0.

Properties of values

If

p

is a value on a vector space

X

then:

An invariant pseudometric that doesn't induce a vector topology

Let

X

be a non-trivial (i.e.

X\{0\}

) 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)=1forallx,y\inX

such that

xy.

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

l{N}

of subsets of a vector space is called additive if for every

N\inl{N},

there exists some

U\inl{N}

such that

U+U\subseteqN.

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\bull=\left(n1,\ldots,nk\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\geq0

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\bull=\left(n1,\ldots,nk\right)

consists of distinct positive integers then

\sum

U
n\bull

\subseteq

U
-1+min\left(n\bull\right)

.

It will now be shown by induction on

k

that if

n\bull=\left(n1,\ldots,nk\right)

consists of non-negative integers such that

\sum

-n\bull
2

\leq2-

for some integer

M\geq0

then

\sum

U
n\bull

\subseteqUM.

This is clearly true for

k=1

and

k=2

so assume that

k>2,

which implies that all

ni

are positive. If all

ni

are distinct then this step is done, and otherwise pick distinct indices

i<j

such that

ni=nj

and construct

m\bull=\left(m1,\ldots,mk-1\right)

from

n\bull

by replacing each

ni

with

ni-1

and deleting the

jth

element of

n\bull

(all other elements of

n\bull

are transferred to

m\bull

unchanged). Observe that

\sum

-n\bull
2

=\sum

-m\bull
2
and

\sum

U
n\bull

\subseteq\sum

U
m\bull
(because
U
ni

+

U
nj

\subseteq

U
ni-1
) so by appealing to the inductive hypothesis we conclude that

\sum

U
n\bull

\subseteq\sum

U
m\bull

\subseteqUM,

as desired.

It is clear that

f(0)=0

and that

0\leqf\leq1

so to prove that

f

is subadditive, it suffices to prove that

f(x+y)\leqf(x)+f(y)

when

x,y\inX

are such that

f(x)+f(y)<1,

which implies that

x,y\inU0.

This is an exercise. If all

Ui

are symmetric then

x\in\sum

U
n\bull
if and only if

-x\in\sum

U
n\bull
from which it follows that

f(-x)\leqf(x)

and

f(-x)\geqf(x).

If all

Ui

are balanced then the inequality

f(sx)\leqf(x)

for all unit scalars

s

such that

|s|\leq1

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

Ui

are neighborhoods of the origin then for any real

r>0,

pick an integer

M>1

such that

2-M<r

so that

x\inUM

implies

f(x)\leq2-M<r.

If the set of all

Ui

form basis of balanced neighborhoods of the origin then it may be shown that for any

n>1,

there exists some

0<r\leq2-n

such that

f(x)<r

implies

x\inUn.

\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\bull=\left(xi\right)

infty
i=1
in

X

and all convergent sequences of scalars

s\bull=\left(si\right)

infty
i=1
":
  1. Continuity of multiplication: if

    s

    is a scalar and

    x\inX

    are such that

    p\left(xi-x\right)\to0

    and

    s\bull\tos,

    then

    p\left(sixi-sx\right)\to0.

  2. Both of the conditions:
    • if

    s\bull\to0

    and if

    x\inX

    is such that

    p\left(xi-x\right)\to0

    then

    p\left(sixi\right)\to0

    ;
    • if

    p\left(x\bull\right)\to0

    then

    p\left(sxi\right)\to0

    for every scalar

    s.

  3. Both of the conditions:
    • if

    p\left(x\bull\right)\to0

    and

    s\bull\tos

    for some scalar

    s

    then

    p\left(sixi\right)\to0

    ;
    • if

    s\bull\to0

    then

    p\left(six\right)\to0forallx\inX.

  4. Separate continuity:
    • if

    s\bull\tos

    for some scalar

    s

    then

    p\left(sxi-sx\right)\to0

    for every

    x\inX

    ;
    • if

    s

    is a scalar,

    x\inX,

    and

    p\left(xi-x\right)\to0

    then

    p\left(sxi-sx\right)\to0

    .

A paranorm is called total if in addition it satisfies:

Properties of paranorms

If

p

is a paranorm on a vector space

X

then the map

d:X x 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:

Examples of paranorms

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\to\Reals

with the following four properties:
  1. Non-negative:

    p\geq0.

  2. Subadditive:

    p(x+y)\leqp(x)+p(y)

    for all

    x,y\inX

  3. Balanced:

    p(ax)\leqp(x)

    for

    x\inX

    all scalars

    a

    satisfying

    |a|\leq1;

    • This condition guarantees that each set of the form

    \{z\inX:p(z)\leqr\}

    or

    \{z\inX:p(z)<r\}

    for some

    r\geq0

    is a balanced set.
  4. For every

    x\inX,

    p\left(\tfrac{1}{n}x\right)\to0

    as

    n\toinfty

    • The sequence
    infty
    \left(\tfrac{1}{n}\right)
    n=1
    can be replaced by any positive sequence converging to the zero.

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

  1. Total/Positive definite:

    p(x)=0

    implies

    x=0.

An F-seminorm is called monotone if it satisfies:

  1. Monotone:

    p(rx)<p(sx)

    for all non-zero

    x\inX

    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\toZ

is called an isometric embedding if

q(f(x)-f(y))=p(x,y)forallx,y\inX.

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

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\inX.

Topology induced by a family of F-seminorms

Suppose that

l{L}

is a non-empty collection of F-seminorms on a vector space

X

and for any finite subset

l{F}\subseteql{L}

and any

r>0,

let U_ := \bigcap_ \.

The set

\left\{Ul{F,r}~:~r>0,l{F}\subseteql{L},l{F}finite\right\}

forms a filter base on

X

that also forms a neighborhood basis at the origin for a vector topology on

X

denoted by

\taul{L

}. Each

Ul{F,r}

is a balanced and absorbing subset of

X.

These sets satisfyU_ + U_ \subseteq U_.

Fréchet combination

Suppose that

p\bull=\left(pi\right)

infty
i=1
is a family of non-negative subadditive functions on a vector space

X.

The Fréchet combination of

p\bull

is defined to be the real-valued map p(x) := \sum_^ \frac.

As an F-seminorm

Assume that

p\bull=\left(pi\right)

infty
i=1
is an increasing sequence of seminorms on

X

and let

p

be the Fréchet combination of

p\bull.

Then

p

is an F-seminorm on

X

that induces the same locally convex topology as the family

p\bull

of seminorms.

Since

p\bull=\left(pi\right)

infty
i=1
is increasing, a basis of open neighborhoods of the origin consists of all sets of the form

\left\{x\inX~:~pi(x)<r\right\}

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

pi

is a paranorm then so is

p

and moreover,

p

induces the same topology on

X

as the family

p\bull

of paranorms. This is also true of the following paranorms on

X

:

Generalization

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

A is a continuous non-negative non-decreasing map

R:[0,infty)\to[0,infty)

that has a bounded range, is subadditive (meaning that

R(s+t)\leqR(s)+R(t)

for all

s,t\geq0

), and satisfies

R(s)=0

if and only if

s=0.

Examples of bounded remetrization functions include

\arctant,

\tanht,

t\mapstomin\{t,1\},

and

t\mapsto

t
1+t

.

If

d

is a pseudometric (respectively, metric) on

X

and

R

is a bounded remetrization function then

R\circd

is a bounded pseudometric (respectively, bounded metric) on

X

that is uniformly equivalent to

d.

Suppose that

p\bull=\left(pi\right)

infty
i=1
is a family of non-negative F-seminorm on a vector space

X,

R

is a bounded remetrization function, and

r\bull=\left(ri\right)

infty
i=1
is a sequence of positive real numbers whose sum is finite. Then p(x) := \sum_^\infty r_i R\left(p_i(x)\right)defines a bounded F-seminorm that is uniformly equivalent to the

p\bull.

It has the property that for any net

x\bull=\left(xa\right)a

in

X,

p\left(x\bull\right)\to0

if and only if

pi\left(x\bull\right)\to0

for all

i.

p

is an F-norm if and only if the

p\bull

separate points on

X.

Characterizations

Of (pseudo)metrics induced by (semi)norms

A pseudometric (resp. metric)

d

is induced by a seminorm (resp. norm) on a vector space

X

if and only if

d

is translation invariant and absolutely homogeneous, which means that for all scalars

s

and all

x,y\inX,

in which case the function defined by

p(x):=d(x,0)

is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by

p

is equal to

d.

Of pseudometrizable TVS

If

(X,\tau)

is a topological vector space (TVS) (where note in particular that

\tau

is assumed to be a vector topology) then the following are equivalent:
  1. X

    is pseudometrizable (i.e. the vector topology

    \tau

    is induced by a pseudometric on

    X

    ).
  2. X

    has a countable neighborhood base at the origin.
  3. The topology on

    X

    is induced by a translation-invariant pseudometric on

    X.

  4. The topology on

    X

    is induced by an F-seminorm.
  5. The topology on

    X

    is induced by a paranorm.

Of metrizable TVS

If

(X,\tau)

is a TVS then the following are equivalent:
  1. X

    is metrizable.
  2. X

    is Hausdorff and pseudometrizable.
  3. X

    is Hausdorff and has a countable neighborhood base at the origin.
  4. The topology on

    X

    is induced by a translation-invariant metric on

    X.

  5. The topology on

    X

    is induced by an F-norm.
  6. The topology on

    X

    is induced by a monotone F-norm.
  7. The topology on

    X

    is induced by a total paranorm.

Of locally convex pseudometrizable TVS

If

(X,\tau)

is TVS then the following are equivalent:
  1. X

    is locally convex and pseudometrizable.
  2. X

    has a countable neighborhood base at the origin consisting of convex sets.
  3. The topology of

    X

    is induced by a countable family of (continuous) seminorms.
  4. The topology of

    X

    is induced by a countable increasing sequence of (continuous) seminorms

    \left(pi\right)

    infty
    i=1
    (increasing means that for all

    i,

    pi\geqpi+1.

  5. The topology of

    X

    is induced by an F-seminorm of the form: p(x) = \sum_^ 2^ \operatorname p_n(x)where

    \left(pi\right)

    infty
    i=1
    are (continuous) seminorms on

    X.

Quotients

Let

M

be a vector subspace of a topological vector space

(X,\tau).

Examples and sufficient conditions

(X,p)

is pseudometrizable with a canonical pseudometric given by

d(x,y):=p(x-y)

for all

x,y\inX.

.
  • If

    (X,d)

    is pseudometric TVS with a translation invariant pseudometric

    d,

    then

    p(x):=d(x,0)

    defines a paranorm. However, if

    d

    is a translation invariant pseudometric on the vector space

    X

    (without the addition condition that

    (X,d)

    is), then

    d

    need not be either an F-seminorm nor a paranorm.
  • If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.
  • If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.
  • Suppose

    X

    is either a DF-space or an LM-space. If

    X

    is a sequential space then it is either metrizable or else a Montel DF-space.
  • If

    X

    is Hausdorff locally convex TVS then

    X

    with the strong topology,

    \left(X,b\left(X,X\prime\right)\right),

    is metrizable if and only if there exists a countable set

    l{B}

    of bounded subsets of

    X

    such that every bounded subset of

    X

    is contained in some element of

    l{B}.

    \prime
    X
    b
    of a metrizable locally convex space (such as a Fréchet space[1])

    X

    is a DF-space. The strong dual of a DF-space is a Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space. The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.If

    X

    is a metrizable locally convex space then its strong dual
    \prime
    X
    b
    has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.

    Normability

    A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable. Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is normable must be infinite dimensional.

    If

    M

    is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then

    M

    is normable.

    If

    X

    is a Hausdorff locally convex space then the following are equivalent:

    1. X

      is normable.
    2. X

      has a (von Neumann) bounded neighborhood of the origin.
    3. the strong dual space
      \prime
      X
      b
      of

      X

      is normable.

    and if this locally convex space

    X

    is also metrizable, then the following may be appended to this list:

    1. the strong dual space of

      X

      is metrizable.
    2. the strong dual space of

      X

      is a Fréchet–Urysohn locally convex space.[1]

    In particular, if a metrizable locally convex space

    X

    (such as a Fréchet space) is normable then its strong dual space
    \prime
    X
    b
    is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space
    \prime
    X
    b
    is also neither metrizable nor normable.

    Another consequence of this is that if

    X

    is a reflexive locally convex TVS whose strong dual
    \prime
    X
    b
    is metrizable then
    \prime
    X
    b
    is necessarily a reflexive Fréchet space,

    X

    is a DF-space, both

    X

    and
    \prime
    X
    b
    are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover,
    \prime
    X
    b
    is normable if and only if

    X

    is normable if and only if

    X

    is Fréchet–Urysohn if and only if

    X

    is metrizable. In particular, such a space

    X

    is either a Banach space or else it is not even a Fréchet–Urysohn space.

    Metrically bounded sets and bounded sets

    Suppose that

    (X,d)

    is a pseudometric space and

    B\subseteqX.

    The set

    B

    is metrically bounded or

    d

    -bounded
    if there exists a real number

    R>0

    such that

    d(x,y)\leqR

    for all

    x,y\inB

    ; the smallest such

    R

    is then called the diameter or

    d

    -diameter
    of

    B.

    If

    B

    is bounded in a pseudometrizable TVS

    X

    then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.

    Properties of pseudometrizable TVS

    Completeness

    See main article: Complete topological vector space.

    Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If

    X

    is a metrizable TVS and

    d

    is a metric that defines

    X

    's topology, then its possible that

    X

    is complete as a TVS (i.e. relative to its uniformity) but the metric

    d

    is a complete metric (such metrics exist even for

    X=\R

    ). Thus, if

    X

    is a TVS whose topology is induced by a pseudometric

    d,

    then the notion of completeness of

    X

    (as a TVS) and the notion of completeness of the pseudometric space

    (X,d)

    are not always equivalent. The next theorem gives a condition for when they are equivalent:

    If

    M

    is a closed vector subspace of a complete pseudometrizable TVS

    X,

    then the quotient space

    X/M

    is complete. If

    M

    is a vector subspace of a metrizable TVS

    X

    and if the quotient space

    X/M

    is complete then so is

    X.

    If

    X

    is not complete then

    M:=X,

    but not complete, vector subspace of

    X.

    A Baire separable topological group is metrizable if and only if it is cosmic.[2]

    Subsets and subsequences

    Generalized series

    As described in this article's section on generalized series, for any

    I

    -indexed family family

    \left(ri\right)i

    of vectors from a TVS

    X,

    it is possible to define their sum

    style\sum\limitsiri

    as the limit of the net of finite partial sums

    F\in\operatorname{FiniteSubsets}(I)\mapstostyle\sum\limitsiri

    where the domain

    \operatorname{FiniteSubsets}(I)

    is directed by

    \subseteq.

    If

    I=\N

    and

    X=\Reals,

    for instance, then the generalized series

    style\sum\limitsiri

    converges if and only if
    infty
    style\sum\limits
    i=1

    ri

    converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series

    style\sum\limitsiri

    converges in a metrizable TVS, then the set

    \left\{i\inI:ri0\right\}

    is necessarily countable (that is, either finite or countably infinite); in other words, all but at most countably many

    ri

    will be zero and so this generalized series

    style\sum\limitsiri~=~style\sum\limits\stackrel{i{ri0}}ri

    is actually a sum of at most countably many non-zero terms.

    Linear maps

    If

    X

    is a pseudometrizable TVS and

    A

    maps bounded subsets of

    X

    to bounded subsets of

    Y,

    then

    A

    is continuous. Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.

    If

    F:X\toY

    is a linear map between TVSs and

    X

    is metrizable then the following are equivalent:

    1. F

      is continuous;
    2. F

      is a (locally) bounded map (that is,

      F

      maps (von Neumann) bounded subsets of

      X

      to bounded subsets of

      Y

      );
    3. F

      is sequentially continuous;
    4. the image under

      F

      of every null sequence in

      X

      is a bounded set where by definition, a is a sequence that converges to the origin.
    5. F

      maps null sequences to null sequences;

    Open and almost open maps

    Theorem: If

    X

    is a complete pseudometrizable TVS,

    Y

    is a Hausdorff TVS, and

    T:X\toY

    is a closed and almost open linear surjection, then

    T

    is an open map.

    Theorem: If

    T:X\toY

    is a surjective linear operator from a locally convex space

    X

    onto a barrelled space

    Y

    (e.g. every complete pseudometrizable space is barrelled) then

    T

    is almost open.

    Theorem: If

    T:X\toY

    is a surjective linear operator from a TVS

    X

    onto a Baire space

    Y

    then

    T

    is almost open.

    Theorem: Suppose

    T:X\toY

    is a continuous linear operator from a complete pseudometrizable TVS

    X

    into a Hausdorff TVS

    Y.

    If the image of

    T

    is non-meager in

    Y

    then

    T:X\toY

    is a surjective open map and

    Y

    is a complete metrizable space.

    Hahn-Banach extension property

    See main article: Hahn-Banach theorem.

    A vector subspace

    M

    of a TVS

    X

    has the extension property if any continuous linear functional on

    M

    can be extended to a continuous linear functional on

    X.

    Say that a TVS

    X

    has the Hahn-Banach extension property (HBEP) if every vector subspace of

    X

    has the extension property.

    The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

    If a vector space

    X

    has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.

    Notes

    Proofs

    Bibliography

    Notes and References

    1. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
    2. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)