Rådström's embedding theorem

In functional analysis, Rådström's embedding theorem is a result related to the set of compact and convex subsets of a normed vector space. It states that such sets can be isometrically embededded into a convex cone in another normed vector space.

The theorem is an important result in that it shows that this family of sets has natural linear and metric structures, which allows for simpler algebraic manipulations via the embedding. It was first proven by Hans Rådström in 1952.^[1]

An extension of Rådström's result to locally convex topological vector spaces, known as the Hörmander embedding theorem, was proven by Lars Hörmander in 1954.^[2]

For any normed vector space ${\displaystyle (X,||\cdot ||)}$, let ${\displaystyle CK(X)}$ be the set of all its convex and compact subsets. We can endow ${\displaystyle CK(X)}$ with a metric structure given by the Hausdorff metric

${\displaystyle d_{H}(A,B)=\sup _{x\in X}|d(x,A)-d(x,B)|,}$

where ${\displaystyle d}$ is the metric over ${\displaystyle X}$ induced by the norm ${\displaystyle ||\cdot ||}$, and ${\displaystyle d(x,Y)=\inf _{y\in Y}d(x,y)}$ is the distance from ${\displaystyle x}$ to a set ${\displaystyle Y\subseteq X}$. It is well known that ${\displaystyle (CK(X),d_{H})}$ forms a metric space of its own right.^[3]

The main version of Rådström's theorem reads as follows:^[1][4]

Theorem (Rådström, 1952): Let ${\displaystyle (X,||\cdot ||)}$ be a normed space. Then there exists a normed space ${\displaystyle (Y,||\cdot ||_{Y})}$ such that the space ${\displaystyle (CK(X),d_{H})}$ can be isometrically embedded into a convex cone ${\displaystyle C\subseteq Y}$. Furthermore, it is possible to construct a "minimal" ${\displaystyle Y}$ for which this holds.^{[nb 1]}

It is possible to generalize Rådström's theorem to locally convex topological vector spaces (LCTVS). This is done via Hörmander's embedding theorem, proven by Lars Hörmander in 1954.^[2] Hörmander's theorem explicitly constructs an embedding via support functions of closed, convex, (strongly) bounded sets.

Theorem (Hörmander, 1954): let ${\displaystyle X}$ be a real LCTVS, and ${\displaystyle CB(X)}$ the set of all its closed, convex, (strongly) bounded subsets. Denoting by ${\displaystyle X^{*}}$ the dual space of ${\displaystyle X}$, consider the closed unit ball ${\displaystyle {\mathcal {B}}^{*}\subseteq X^{*}}$, and the Banach space ${\displaystyle {\mathcal {C}}_{b}({\mathcal {B}}^{*})}$ of all continuous, bounded functions ${\displaystyle h:B^{*}\to \mathbb {R} }$. For each ${\displaystyle A\in CB(X)}$, let ${\displaystyle s_{A}\in {\mathcal {C}}_{b}({\mathcal {B}}^{*})}$ defined by

${\displaystyle s_{A}(x)=\sup _{y\in A}\langle x,y\rangle }$

be its support function. Then the mapping ${\displaystyle H:CB(X)\to {\mathcal {C}}_{b}({\mathcal {B}}^{*})}$ given by

${\displaystyle A\mapsto s_{A}}$

is an isometric embedding into a convex cone in ${\displaystyle {\mathcal {C}}_{b}({\mathcal {B}}^{*})}$. Moreover, by the properties of support functions such embedding is also linear when ${\displaystyle CB(X)}$ is endowed with a vector-space structure given by the Minkowski sum and scalar multiplication.^[4]

The theorem can be used to define the integral of a set-valued function (or correspondence) via Debreu's integral.^[5][6] This has applications, for example, in the theory of random compact sets.^[7][8]