  We presented a systematic treatment of a Hilbert criterion for stability
theory for an action of a real reductive group $G$ on a real submanifold $X$ of
a K\"ahler manifold $Z$. More precisely, we suppose the action of a compact
connected Lie group $U$ with Lie algebra $\mathfrak{u}$ extends holomorphically
to an action of the complexified group $U^{\mathbb C}$ and that the $U$-action
on $Z$ is Hamiltonian. If $G\subset U^{\mathbb C}$ is closed and compatible,
there is a corresponding gradient map $\mu_\mathfrak{p} : X\longrightarrow
\mathfrak{p}$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartan
decomposition of the Lie algebra of $G$. The concept of energy complete action
of $G$ on $X$ is introduced. For such actions, one can characterize stability,
semistability and polystability of a point by a numerical criteria using a
$G$-equivariant function associated with a gradient map, called maximal weight
function. We also prove the classical Hilbert-Mumford criteria for semistabilty
and polystability conditions.
  We thank the anonymous referee for carefully reading our paper and for giving
such constructive comments which substantially helped improving the quality of
the paper.
