Let
S
=
{
x
∈
R
n
∣
f
1
(
x
)
≥
0
,
…
,
f
s
(
x
)
≥
0
}
\mathcal {S}=\{x\in \mathbb {R}^n\mid f_1(x)\geq 0,\ldots ,f_s(x)\geq 0\}
be a basic closed semi-algebraic set in
R
n
\mathbb {R}^n
and let
P
O
(
f
1
,
…
,
f
s
)
\mathrm {PO}(f_1,\ldots ,f_s)
be the corresponding preordering in
R
[
X
1
,
…
,
X
n
]
\mathbb {R}[X_1,\ldots ,X_n]
. We examine for which polynomials
f
f
there exist identities
\[
f
+
ε
q
∈
P
O
(
f
1
,
…
,
f
s
)
for all
ε
>
0.
f+\varepsilon q\in \mathrm {PO}(f_1,\ldots ,f_s) \mbox { for all } \varepsilon >0.
\]
These are precisely the elements of the sequential closure of
P
O
(
f
1
,
…
,
f
s
)
\mathrm {PO}(f_1,\ldots ,f_s)
with respect to the finest locally convex topology. We solve the open problem from Kuhlmann, Marshall, and Schwartz (2002, 2005), whether this equals the double dual cone
\[
P
O
(
f
1
,
…
,
f
s
)
∨
∨
,
\mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee },
\]
by providing a counterexample. We then prove a theorem that allows us to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial
f
f
that is nonnegative on
S
\mathcal {S}
admits such representations, or at least the polynomials from
P
O
(
f
1
,
…
,
f
s
)
∨
∨
\mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee }
do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.