  In the present paper we suggest a construction of symmetric functionals on a
large class of symmetric spaces over a semifinite von Neumann algebra. This
approach establishes a bijection between the symmetric functionals on symmetric
spaces and shift-invariant functionals on the space of bounded sequences. It
allows to obtain a bijection between the classes of all continuous symmetric
functionals on different symmetric spaces. Notably, we show that this mapping
is not bijective on the class of all Dixmier traces. As an application of our
results we prove an extension of the Connes trace formula for a wide class of
operators and symmetric functionals.
