Normal cone (functional analysis)

In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at the origin in a topological vector space X {\displaystyle X} such that 0 ∈ C {\displaystyle 0\in C} and if U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, then C {\displaystyle C} is called normal if U = [ U ] C , {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C},} where [ U ] C := { [ U ] C : U ∈ U } {\displaystyle \left[{\mathcal {U}}\right]_{C}:=\left\{[U]_{C}:U\in {\mathcal {U}}\right\}} and where for any subset S ⊆ X , {\displaystyle S\subseteq X,} [ S ] C := ( S + C ) ∩ ( S − C ) {\displaystyle [S]_{C}:=(S+C)\cap (S-C)} is the C {\displaystyle C}-saturatation of S . {\displaystyle S.}

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

Characterizations

If C {\displaystyle C} is a cone in a TVS X {\displaystyle X} then for any subset S ⊆ X {\displaystyle S\subseteq X} let [ S ] C := ( S + C ) ∩ ( S − C ) {\displaystyle [S]_{C}:=\left(S+C\right)\cap \left(S-C\right)} be the C {\displaystyle C}-saturated hull of S ⊆ X {\displaystyle S\subseteq X} and for any collection S {\displaystyle {\mathcal {S}}} of subsets of X {\displaystyle X} let [ S ] C := { [ S ] C : S ∈ S } . {\displaystyle \left[{\mathcal {S}}\right]_{C}:=\left\{\left[S\right]_{C}:S\in {\mathcal {S}}\right\}.} If C {\displaystyle C} is a cone in a TVS X {\displaystyle X} then C {\displaystyle C} is normal if U = [ U ] C , {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C},} where U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin.

If T {\displaystyle {\mathcal {T}}} is a collection of subsets of X {\displaystyle X} and if F {\displaystyle {\mathcal {F}}} is a subset of T {\displaystyle {\mathcal {T}}} then F {\displaystyle {\mathcal {F}}} is a fundamental subfamily of T {\displaystyle {\mathcal {T}}} if every T ∈ T {\displaystyle T\in {\mathcal {T}}} is contained as a subset of some element of F . {\displaystyle {\mathcal {F}}.} If G {\displaystyle {\mathcal {G}}} is a family of subsets of a TVS X {\displaystyle X} then a cone C {\displaystyle C} in X {\displaystyle X} is called a G {\displaystyle {\mathcal {G}}}-cone if { [ G ] C ¯ : G ∈ G } {\displaystyle \left\{{\overline {\left[G\right]_{C}}}:G\in {\mathcal {G}}\right\}} is a fundamental subfamily of G {\displaystyle {\mathcal {G}}} and C {\displaystyle C} is a strict G {\displaystyle {\mathcal {G}}}-cone if { [ G ] C : G ∈ G } {\displaystyle \left\{\left[G\right]_{C}:G\in {\mathcal {G}}\right\}} is a fundamental subfamily of G . {\displaystyle {\mathcal {G}}.} Let B {\displaystyle {\mathcal {B}}} denote the family of all bounded subsets of X . {\displaystyle X.}

If C {\displaystyle C} is a cone in a TVS X {\displaystyle X} (over the real or complex numbers), then the following are equivalent:

C {\displaystyle C} is a normal cone.
For every filter F {\displaystyle {\mathcal {F}}} in X , {\displaystyle X,} if lim F = 0 {\displaystyle \lim {\mathcal {F}}=0} then lim [ F ] C = 0. {\displaystyle \lim \left[{\mathcal {F}}\right]_{C}=0.}
There exists a neighborhood base G {\displaystyle {\mathcal {G}}} in X {\displaystyle X} such that B ∈ G {\displaystyle B\in {\mathcal {G}}} implies [ B ∩ C ] C ⊆ B . {\displaystyle \left[B\cap C\right]_{C}\subseteq B.}

and if X {\displaystyle X} is a vector space over the reals then we may add to this list:

There exists a neighborhood base at the origin consisting of convex, balanced, C {\displaystyle C}-saturated sets.
There exists a generating family P {\displaystyle {\mathcal {P}}} of semi-norms on X {\displaystyle X} such that p ( x ) ≤ p ( x + y ) {\displaystyle p(x)\leq p(x+y)} for all x , y ∈ C {\displaystyle x,y\in C} and p ∈ P . {\displaystyle p\in {\mathcal {P}}.}

and if X {\displaystyle X} is a locally convex space and if the dual cone of C {\displaystyle C} is denoted by X ′ {\displaystyle X^{\prime }} then we may add to this list:

For any equicontinuous subset S ⊆ X ′ , {\displaystyle S\subseteq X^{\prime },} there exists an equicontiuous B ⊆ C ′ {\displaystyle B\subseteq C^{\prime }} such that S ⊆ B − B . {\displaystyle S\subseteq B-B.}
The topology of X {\displaystyle X} is the topology of uniform convergence on the equicontinuous subsets of C ′ . {\displaystyle C^{\prime }.}

and if X {\displaystyle X} is an infrabarreled locally convex space and if B ′ {\displaystyle {\mathcal {B}}^{\prime }} is the family of all strongly bounded subsets of X ′ {\displaystyle X^{\prime }} then we may add to this list:

The topology of X {\displaystyle X} is the topology of uniform convergence on strongly bounded subsets of C ′ . {\displaystyle C^{\prime }.}
C ′ {\displaystyle C^{\prime }} is a B ′ {\displaystyle {\mathcal {B}}^{\prime }}-cone in X ′ . {\displaystyle X^{\prime }.} this means that the family { [ B ′ ] C ¯ : B ′ ∈ B ′ } {\displaystyle \left\{{\overline {\left[B^{\prime }\right]_{C}}}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}} is a fundamental subfamily of B ′ . {\displaystyle {\mathcal {B}}^{\prime }.}
C ′ {\displaystyle C^{\prime }} is a strict B ′ {\displaystyle {\mathcal {B}}^{\prime }}-cone in X ′ . {\displaystyle X^{\prime }.} this means that the family { [ B ′ ] C : B ′ ∈ B ′ } {\displaystyle \left\{\left[B^{\prime }\right]_{C}:B^{\prime }\in {\mathcal {B}}^{\prime }\right\}} is a fundamental subfamily of B ′ . {\displaystyle {\mathcal {B}}^{\prime }.}

and if X {\displaystyle X} is an ordered locally convex TVS over the reals whose positive cone is C , {\displaystyle C,} then we may add to this list:

there exists a Hausdorff locally compact topological space S {\displaystyle S} such that X {\displaystyle X} is isomorphic (as an ordered TVS) with a subspace of R ( S ) , {\displaystyle R(S),} where R ( S ) {\displaystyle R(S)} is the space of all real-valued continuous functions on X {\displaystyle X} under the topology of compact convergence.

If X {\displaystyle X} is a locally convex TVS, C {\displaystyle C} is a cone in X {\displaystyle X} with dual cone C ′ ⊆ X ′ , {\displaystyle C^{\prime }\subseteq X^{\prime },} and G {\displaystyle {\mathcal {G}}} is a saturated family of weakly bounded subsets of X ′ , {\displaystyle X^{\prime },} then

if C ′ {\displaystyle C^{\prime }} is a G {\displaystyle {\mathcal {G}}}-cone then C {\displaystyle C} is a normal cone for the G {\displaystyle {\mathcal {G}}}-topology on X {\displaystyle X};
if C {\displaystyle C} is a normal cone for a G {\displaystyle {\mathcal {G}}}-topology on X {\displaystyle X} consistent with ⟨ X , X ′ ⟩ {\displaystyle \left\langle X,X^{\prime }\right\rangle } then C ′ {\displaystyle C^{\prime }} is a strict G {\displaystyle {\mathcal {G}}}-cone in X ′ . {\displaystyle X^{\prime }.}

If X {\displaystyle X} is a Banach space, C {\displaystyle C} is a closed cone in X , {\displaystyle X,}, and B ′ {\displaystyle {\mathcal {B}}^{\prime }} is the family of all bounded subsets of X b ′ {\displaystyle X_{b}^{\prime }} then the dual cone C ′ {\displaystyle C^{\prime }} is normal in X b ′ {\displaystyle X_{b}^{\prime }} if and only if C {\displaystyle C} is a strict B {\displaystyle {\mathcal {B}}}-cone.

If X {\displaystyle X} is a Banach space and C {\displaystyle C} is a cone in X {\displaystyle X} then the following are equivalent:

C {\displaystyle C} is a B {\displaystyle {\mathcal {B}}}-cone in X {\displaystyle X};
X = C ¯ − C ¯ {\displaystyle X={\overline {C}}-{\overline {C}}};
C ¯ {\displaystyle {\overline {C}}} is a strict B {\displaystyle {\mathcal {B}}}-cone in X . {\displaystyle X.}

Ordered topological vector spaces

Suppose L {\displaystyle L} is an ordered topological vector space. That is, L {\displaystyle L} is a topological vector space, and we define x ≥ y {\displaystyle x\geq y} whenever x − y {\displaystyle x-y} lies in the cone L + {\displaystyle L_{+}}. The following statements are equivalent:

The cone L + {\displaystyle L_{+}} is normal;
The normed space L {\displaystyle L} admits an equivalent monotone norm;
There exists a constant c > 0 {\displaystyle c>0} such that a ≤ x ≤ b {\displaystyle a\leq x\leq b} implies ‖ x ‖ ≤ c max { ‖ a ‖ , ‖ b ‖ } {\displaystyle \lVert x\rVert \leq c\max\{\lVert a\rVert ,\lVert b\rVert \}};
The full hull [ U ] = ( U + L + ) ∩ ( U − L + ) {\displaystyle [U]=(U+L_{+})\cap (U-L_{+})} of the closed unit ball U {\displaystyle U} of L {\displaystyle L} is norm bounded;
There is a constant c > 0 {\displaystyle c>0} such that 0 ≤ x ≤ y {\displaystyle 0\leq x\leq y} implies ‖ x ‖ ≤ c ‖ y ‖ {\displaystyle \lVert x\rVert \leq c\lVert y\rVert }.

Properties

If X {\displaystyle X} is a Hausdorff TVS then every normal cone in X {\displaystyle X} is a proper cone.
If X {\displaystyle X} is a normable space and if C {\displaystyle C} is a normal cone in X {\displaystyle X} then X ′ = C ′ − C ′ . {\displaystyle X^{\prime }=C^{\prime }-C^{\prime }.}
Suppose that the positive cone of an ordered locally convex TVS X {\displaystyle X} is weakly normal in X {\displaystyle X} and that Y {\displaystyle Y} is an ordered locally convex TVS with positive cone D . {\displaystyle D.} If Y = D − D {\displaystyle Y=D-D} then H − H {\displaystyle H-H} is dense in L s ( X ; Y ) {\displaystyle L_{s}(X;Y)} where H {\displaystyle H} is the canonical positive cone of L ( X ; Y ) {\displaystyle L(X;Y)} and L s ( X ; Y ) {\displaystyle L_{s}(X;Y)} is the space L ( X ; Y ) {\displaystyle L(X;Y)} with the topology of simple convergence. If G {\displaystyle {\mathcal {G}}} is a family of bounded subsets of X , {\displaystyle X,} then there are apparently no simple conditions guaranteeing that H {\displaystyle H} is a T {\displaystyle {\mathcal {T}}}-cone in L G ( X ; Y ) , {\displaystyle L_{\mathcal {G}}(X;Y),} even for the most common types of families T {\displaystyle {\mathcal {T}}} of bounded subsets of L G ( X ; Y ) {\displaystyle L_{\mathcal {G}}(X;Y)} (except for very special cases).

Sufficient conditions

If the topology on X {\displaystyle X} is locally convex then the closure of a normal cone is a normal cone.

Suppose that { X α : α ∈ A } {\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}} is a family of locally convex TVSs and that C α {\displaystyle C_{\alpha }} is a cone in X α . {\displaystyle X_{\alpha }.} If X := ⨁ α X α {\displaystyle X:=\bigoplus _{\alpha }X_{\alpha }} is the locally convex direct sum then the cone C := ⨁ α C α {\displaystyle C:=\bigoplus _{\alpha }C_{\alpha }} is a normal cone in X {\displaystyle X} if and only if each X α {\displaystyle X_{\alpha }} is normal in X α . {\displaystyle X_{\alpha }.}

If X {\displaystyle X} is a locally convex space then the closure of a normal cone is a normal cone.

If C {\displaystyle C} is a cone in a locally convex TVS X {\displaystyle X} and if C ′ {\displaystyle C^{\prime }} is the dual cone of C , {\displaystyle C,} then X ′ = C ′ − C ′ {\displaystyle X^{\prime }=C^{\prime }-C^{\prime }} if and only if C {\displaystyle 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 {\displaystyle X} and Y {\displaystyle Y} are ordered locally convex TVSs and if G {\displaystyle {\mathcal {G}}} is a family of bounded subsets of X , {\displaystyle X,} then if the positive cone of X {\displaystyle X} is a G {\displaystyle {\mathcal {G}}}-cone in X {\displaystyle X} and if the positive cone of Y {\displaystyle Y} is a normal cone in Y {\displaystyle Y} then the positive cone of L G ( X ; Y ) {\displaystyle L_{\mathcal {G}}(X;Y)} is a normal cone for the G {\displaystyle {\mathcal {G}}}-topology on L ( X ; Y ) . {\displaystyle L(X;Y).}

Bibliography