An H-space is a topological space with a continuous multiplication and an identity element. In this paper X has the homotopy type of a countable CW-complex with integral cohomology of finite type and primitively generated k-cohomology, k a field. The projective n-plane of X is denoted
X
P
(
n
)
XP(n)
. The main results of this paper are: Theorem 1 which states that
H
∗
(
X
P
(
n
)
)
=
N
⊕
S
{H^\ast }(XP(n)) = N \oplus S
where N is a truncated polynomial algebra over k and S is a trivial k-ideal, and Theorem 2 which considers the case
k
=
Z
(
p
)
k = Z(p)
and states that
H
∗
(
X
P
(
n
)
)
=
N
^
⊕
S
^
{H^\ast }(XP(n)) = \hat N \oplus \hat S
where
N
^
\hat N
is a truncated polynomial algebra on generators in even dimensions and S is an A(p)-sub-algebra of
H
∗
(
X
P
(
n
)
)
{H^\ast }(XP(n))
so that an A(p)-algebra structure can be induced on
N
^
\hat N
. These theorems extend results by A. Borel, W. Browder, M. Rothenberg, N. E. Steenrod, and E. Thomas.