An example of a linear functional defined on a dense subspace of the Hardy space
H
1
(
R
n
)
H^1(\mathbb {R}^n)
is constructed. It is shown that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole
H
1
H^1
. Therefore, this shows that in general it is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer which states that quasi-norms corresponding to finite and infinite atomic decompositions in
H
p
H^p
,
0
>
p
≤
1
0>p \le 1
, are not equivalent.