We investigate pointwise nonnegativity as an obstruction to various types of structured completeness in
L
p
(
R
)
L^p(\mathbb {R})
. For example, we prove that if each element of the system
{
f
n
}
n
=
1
∞
⊂
L
p
(
R
)
\{f_n\}_{n=1}^\infty \subset L^p(\mathbb {R})
is pointwise nonnegative, then
{
f
n
}
n
=
1
∞
\{f_n\}_{n=1}^{\infty }
cannot be an unconditional basis or unconditional quasibasis (unconditional Schauder frame) for
L
p
(
R
)
L^p(\mathbb {R})
. In particular, in
L
2
(
R
)
L^2(\mathbb {R})
this precludes the existence of nonnegative Riesz bases and frames. On the other hand, there exist pointwise nonnegative conditional quasibases in
L
p
(
R
)
L^p(\mathbb {R})
, and there also exist pointwise nonnegative exact systems and Markushevich bases in
L
p
(
R
)
L^p(\mathbb {R})
.