Let $X$ be a real analytic paracompact space (with nilpotent elements) then $X$ has a complexification $\tilde{X}$ and there exists a fundamental system of neighbourhoods of $X$ in $\tilde{X}$ that are Stein spaces. It follows that the theorems A and B hold for every coherent sheaf of modules on an analytic real space. Let $V^{m}$ be a complex $m$-dimensional manifold and $\sigma : V^{m} \to V^{m}$ an antiinvolution, then the set $V' = \{ x e V^{m} \mid \sigma(x) = x \}$ is a real $m$-dimensional analytic manifold or the empty set. From these results follow imbedding theorems of real analytic $n$-dimensional space in $\mathrm{R}^{4n+2}$. In the last part it is proved that any real analytic space (without nilpotent elements) admits a (global) decomposition into irreducible components and a normalization.