N.Teleman in 1983 had defined and examined the signature operator on Lipschitz compact manifolds M for L2-differential forms L2(M). The space L2(M) (in opposite to smooth differential forms) is Hilbert, but de-Rham derivative of such differential forms (determined in distributional manner) is not defined on all space L2(M) (but only on some dense subspace). The fundamental observation by N.Teleman is that the Poincaré duality property - which is obtained via algebraic topology methods - is sufficient to otain (in L2-theory) the Hodge theory (i.e. Hodge isomorphism and the strong Hodge decomposition) and the calculation of the signature via the index of the suitable Hirzebruch operator (the convolution L2-argument and the L2-Poincaré Lemma is all what it is needed). In the talk we give algebraic point of view on (abstract) graded Hilbert Hodge spaces with derivative defined on some dense subspace. The Poincaré duality is sufficient to obtain a strong Hodge decomposition theorem and the Hodge isomorphism. Therefore the suitable (abstract) Hirzebruch operator has index equaling to the signature. For non Hilbert case we can do Hilbert completion and extend the derivative in a distributional sense. The diagram joining two Hodge homomorphisms is obtained. The conditions under which all homomorphisms in this diagram are isomorphisms are given. Then the signature can be calculated via Hirzebruch operator for two manners and we obtain the same number. As application we examine completion of the space of differential forms for our four fundamental examples (classical, Lie algebroid, Lusztig, Gromov). The mentioned above conditions are fullfilled, for our four examples we may prove them without elliptic theory by use the Mayer-Vietoris or spectral sequences argument and sheaves argument.