Let
H
H
be an infinite-dimensional Hilbert space of density character
m
\mathfrak {m}
. By representing
H
H
as a module over an appropriate Clifford algebra, it is proved that
H
H
possesses a family
{
A
α
}
α
∈
m
\{A_{\alpha }\}_{\alpha \in \mathfrak {m}}
of proper closed nonzero subspaces such that
d
(
S
A
α
,
S
A
β
)
=
d
(
S
A
α
⊥
,
S
A
β
)
=
d
(
S
A
α
⊥
,
S
A
β
⊥
)
=
2
−
2
(
α
≠
β
)
.
\begin{equation*}d(S_{A_{\alpha }},S_{A_{\beta }})=d(S_{A^{\perp }_{\alpha }},S_{A_{\beta }}) =d(S_{A^{\perp }_{\alpha }},S_{A^{\perp }_{\beta }})=\sqrt {2-\sqrt 2}\qquad (\alpha \ne \beta ).\end{equation*}
Analogous results are proved for
L
p
L_{p}
spaces and for
c
0
(
X
)
c_{0}(X)
and
ℓ
p
(
X
)
\ell _{p}(X)
(
1
≤
p
≤
∞
1 \le p \le \infty
) when
X
X
is an arbitrary nonzero Banach space.