  The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and
Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the
linear isometric structure, the algebraic structure, and the lattice structure,
respectively, of the space $C(X)$. In this paper, it is shown that for rather
general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any
linear bijection $T: A(X) \to A(Y)$ such that $f \geq 0$ if and only if $Tf
\geq 0$ gives rise to a homeomorphism $h: X \to Y$ with which $T$ can be
represented as a weighted composition operator. The three classical results
mentioned above can be derived as corollaries. Generalizations to noncompact
spaces and other function spaces such as spaces of uniformly continuous
functions, Lipschitz functions and differentiable functions are presented.
