  Let $\mathcal{E}$ be a Banach space contained in a Hilbert space
$\mathcal{L}$. Assume that the inclusion is continuous with dense range.
Following the terminology of Gohberg and Zambicki\v{\i}, we say that a bounded
operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with
respect to the inner product of $\mathcal{L}$. By a proper subspace
$\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of
a proper projection. If there exists a proper projection which is also
self-adjoint with respect to the inner product of $\mathcal{L}$, then
$\mathcal{S}$ belongs to a well-known class of subspaces called compatible
subspaces. We find equivalent conditions to describe proper subspaces. Then we
prove a necessary and sufficient condition to ensure that a proper subspace is
compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$
which is also a proper subspace. We give a characterization of the
compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several
examples are provided that illustrate different situations between proper and
compatible subspaces.
