Rådström's embedding theorem

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]