  It is shown that the polar decomposition theorem of operators in (real)
Hilbert spaces gives rise to the known decomposition in boost and spatial
rotation part of any matrix of the orthochronous proper Lorentz group
$SO(1,3)\uparrow$. This result is not trivial because the polar decomposition
theorem is referred to a positive defined scalar product while the
Lorentz-group decomposition theorem deals with the indefinite Lorentz metric. A
generalization to infinite dimensional spaces can be given. It is finally shown
that the polar decomposition of $SL(2,\bC)$ is preserved by the covering
homomorphism of $SL(2,\bC)$ onto $SO(1,3)\spa\uparrow$
