Normal cone (functional analysis) explained

In mathematics, specifically in order theory and functional analysis, if

C

is a cone at the origin in a topological vector space

X

such that

0\inC

and if

l{U}

is the neighborhood filter at the origin, then

C

is called normal if

l{U}=\left[l{U}\right]C,

where

\left[l{U}\right]C:=\left\{[U]C:U\inl{U}\right\}

and where for any subset

S\subseteqX,

[S]C:=(S+C)\cap(S-C)

is the

C

-saturatation of

S.

Normal cones play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Characterizations

If

C

is a cone in a TVS

X

then for any subset

S\subseteqX

let

[S]C:=\left(S+C\right)\cap\left(S-C\right)

be the

C

-saturated hull of

S\subseteqX

and for any collection

l{S}

of subsets of

X

let

\left[l{S}\right]C:=\left\{\left[S\right]C:S\inl{S}\right\}.

If

C

is a cone in a TVS

X

then

C

is normal if

l{U}=\left[l{U}\right]C,

where

l{U}

is the neighborhood filter at the origin.

If

l{T}

is a collection of subsets of

X

and if

l{F}

is a subset of

l{T}

then

l{F}

is a fundamental subfamily of

l{T}

if every

T\inl{T}

is contained as a subset of some element of

l{F}.

If

l{G}

is a family of subsets of a TVS

X

then a cone

C

in

X

is called a

l{G}

-cone if

\left\{\overline{\left[G\right]C}:G\inl{G}\right\}

is a fundamental subfamily of

l{G}

and

C

is a strict

l{G}

-cone if

\left\{\left[G\right]C:G\inl{G}\right\}

is a fundamental subfamily of

l{G}.

Let

l{B}

denote the family of all bounded subsets of

X.

If

C

is a cone in a TVS

X

(over the real or complex numbers), then the following are equivalent:

  1. C

    is a normal cone.
  2. For every filter

    l{F}

    in

    X,

    if

    \liml{F}=0

    then

    \lim\left[l{F}\right]C=0.

  3. There exists a neighborhood base

    l{G}

    in

    X

    such that

    B\inl{G}

    implies

    \left[B\capC\right]C\subseteqB.

and if

X

is a vector space over the reals then we may add to this list:
  1. There exists a neighborhood base at the origin consisting of convex, balanced,

    C

    -saturated     sets.
  2. There exists a generating family

    l{P}

    of semi-norms on

    X

    such that

    p(x)\leqp(x+y)

    for all

    x,y\inC

    and

    p\inl{P}.

and if

X

is a locally convex space and if the dual cone of

C

is denoted by

X\prime

then we may add to this list:
  1. For any equicontinuous subset

    S\subseteqX\prime,

    there exists an equicontiuous

    B\subseteqC\prime

    such that

    S\subseteqB-B.

  2. The topology of

    X

    is the topology of uniform convergence on the equicontinuous subsets of

    C\prime.

and if

X

is an infrabarreled locally convex space and if

l{B}\prime

is the family of all strongly bounded subsets of

X\prime

then we may add to this list:
  1. The topology of

    X

    is the topology of uniform convergence on strongly bounded subsets of

    C\prime.

  2. C\prime

    is a

    l{B}\prime

    -cone in

    X\prime.

    • this means that the family

    \left\{\overline{\left[B\prime\right]C}:B\prime\inl{B}\prime\right\}

    is a fundamental subfamily of

    l{B}\prime.

  3. C\prime

    is a strict

    l{B}\prime

    -cone in

    X\prime.

    • this means that the family

    \left\{\left[B\prime\right]C:B\prime\inl{B}\prime\right\}

    is a fundamental subfamily of

    l{B}\prime.

and if

X

is an ordered locally convex TVS over the reals whose positive cone is

C,

then we may add to this list:
  1. there exists a Hausdorff locally compact topological space

    S

    such that

    X

    is isomorphic (as an ordered TVS) with a subspace of

    R(S),

    where

    R(S)

    is the space of all real-valued continuous functions on

    X

    under the topology of compact convergence.

If

X

is a locally convex TVS,

C

is a cone in

X

with dual cone

C\prime\subseteqX\prime,

and

l{G}

is a saturated family of weakly bounded subsets of

X\prime,

then
  1. if

C\prime

is a

l{G}

-cone then

C

is a normal cone for the

l{G}

-topology on

X

;
  1. if

C

is a normal cone for a

l{G}

-topology on

X

consistent with

\left\langleX,X\prime\right\rangle

then

C\prime

is a strict

l{G}

-cone in

X\prime.

If

X

is a Banach space,

C

is a closed cone in

X,

, and

l{B}\prime

is the family of all bounded subsets of
\prime
X
b
then the dual cone

C\prime

is normal in
\prime
X
b
if and only if

C

is a strict

l{B}

-cone.

If

X

is a Banach space and

C

is a cone in

X

then the following are equivalent:

C

is a

l{B}

-cone in

X

;

X=\overline{C}-\overline{C}

;

\overline{C}

is a strict

l{B}

-cone in

X.

Ordered topological vector spaces

Suppose

L

is an ordered topological vector space. That is,

L

is a topological vector space, and we define

x\geqy

whenever

x-y

lies in the cone

L+

. The following statements are equivalent:[1]
  1. The cone

L+

is normal;
  1. The normed space

L

admits an equivalent monotone norm;
  1. There exists a constant

c>0

such that

a\leqx\leqb

implies

\lVertx\rVert\leqcmax\{\lVerta\rVert,\lVertb\rVert\}

;
  1. The full hull

[U]=(U+L+)\cap(U-L+)

of the closed unit ball

U

of

L

is norm bounded;
  1. There is a constant

c>0

such that

0\leqx\leqy

implies

\lVertx\rVert\leqc\lVerty\rVert

.

Properties

X

is a Hausdorff TVS then every normal cone in

X

is a proper cone.

X

is a normable space and if

C

is a normal cone in

X

then

X\prime=C\prime-C\prime.

X

is weakly normal in

X

and that

Y

is an ordered locally convex TVS with positive cone

D.

If

Y=D-D

then

H-H

is dense in

Ls(X;Y)

where

H

is the canonical positive cone of

L(X;Y)

and

Ls(X;Y)

is the space

L(X;Y)

with the topology of simple convergence.

l{G}

is a family of bounded subsets of

X,

then there are apparently no simple conditions guaranteeing that

H

is a

l{T}

-cone in

Ll{G

}(X; Y), even for the most common types of families

l{T}

of bounded subsets of

Ll{G

}(X; Y) (except for very special cases).

Sufficient conditions

If the topology on

X

is locally convex then the closure of a normal cone is a normal cone.

Suppose that

\left\{X\alpha:\alpha\inA\right\}

is a family of locally convex TVSs and that

C\alpha

is a cone in

X\alpha.

If

X:=oplus\alphaX\alpha

is the locally convex direct sum then the cone

C:=oplus\alphaC\alpha

is a normal cone in

X

if and only if each

X\alpha

is normal in

X\alpha.

If

X

is a locally convex space then the closure of a normal cone is a normal cone.

If

C

is a cone in a locally convex TVS

X

and if

C\prime

is the dual cone of

C,

then

X\prime=C\prime-C\prime

if and only if

C

is weakly normal. Every normal cone in a locally convex TVS is weakly normal. In a normed space, a cone is normal if and only if it is weakly normal.

If

X

and

Y

are ordered locally convex TVSs and if

l{G}

is a family of bounded subsets of

X,

then if the positive cone of

X

is a

l{G}

-cone in

X

and if the positive cone of

Y

is a normal cone in

Y

then the positive cone of

Ll{G

}(X; Y) is a normal cone for the

l{G}

-topology on

L(X;Y).

Notes and References

  1. Book: Aliprantis, Charalambos D. . Cones and duality . 2007 . American Mathematical Society . Rabee Tourky . 978-0-8218-4146-4 . Providence, R.I. . 87808043.

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