Embedding normed linear spaces into C(X)
Abstract
It is well known that every (real or complex) normed linear space
L
is isometrically embeddable into
C(X)
for some compact Hausdorff space
X
. Here
X
is the closed unit ball of
L
∗
(the set of all continuous scalar-valued linear mappings on
L
) endowed with the weak
∗
topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space
X
can indeed be chosen to be the Stone-Cech compactification of
L
∗
∖{0}
, where
L
∗
∖{0}
is endowed with the supremum norm topology.