Abstract
AbstractWe study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space X. Then we turn to the case that $$X=\mathcal {F}(\Omega )$$
X
=
F
(
Ω
)
is a space of scalar-valued functions on a non-empty set $$\Omega $$
Ω
and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there are a locally convex Hausdorff space Y, a map $$\delta :\Omega \rightarrow Y$$
δ
:
Ω
→
Y
and a topological isomorphism $$T:\mathcal {F}(\Omega )\rightarrow Y_{b}'$$
T
:
F
(
Ω
)
→
Y
b
′
such that $$T(f)\circ \delta = f$$
T
(
f
)
∘
δ
=
f
for all $$f\in \mathcal {F}(\Omega )$$
f
∈
F
(
Ω
)
.
Publisher
Springer Science and Business Media LLC