A bounded linear operator

*A*over a Hilbert space

**H**is said to be in the

**trace class**if for some (and hence all) orthonormal bases Ω of

**H**; the sum

**trace**of

*A*, denoted by tr(

*A*) and is independent of the choice of the orthonormal bases.

When **H** is finite-dimensional, then the trace of *A* is just the trace of a matrix and the last property stated above is roughly saying that trace is invariant under similarity.

The trace is a linear functional over the trace class, meaning

*A*,

*B*>=tr(

*AB*

^{*}) is an inner product on the trace class, where the induced norm is called the

**trace norm**.