  For an index set $\Gamma$ and a cardinal number $\kappa$ the
$\Sigma_{\kappa}$-product of real lines $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$
consist of all elements of $\mathbb{R}^{\Gamma}$ with $<\kappa$ nonzero
coordinates. A compact space is $\kappa$-Corson if it can be embedded into
$\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ for some $\Gamma$. We also consider a
class of compact spaces wider than the class of $\omega$-Corson compact spaces,
investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and
Zakrzewski called $NY$ compact spaces. For a Tychonoff space $X$, let
$C_{p}(X)$ be the space of real continuous functions on the space $X$, endowed
with the pointwise convergence topology. We present here a characterisation of
$\kappa$-Corson compact spaces $K$ for regular, uncountable cardinal numbers
$\kappa$ in terms of function spaces $C_{p}(K)$, extending a theorem of Bell
and Marciszewski and a theorem of Pol. We also prove that classes of $NY$
compact spaces and $\omega$-Corson compact spaces $K$ are preserved by linear
homeomorphisms of function spaces $C_{p}(K)$.
