  We study the structure of tensor representations of the classical
infinite-dimensional locally finite Lie algebras $gl_\infty$, $sl_\infty$,
$sp_\infty$ and $so_\infty$. In contrast with the finite-dimensional case,
these tensor representations are not semisimple. We explicitly describe their
Jordan-Holder constituents, socle filtrations, and indecomposable direct
summands.
