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Trace class

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

<math>\sum_{x\in \Omega}<Ax,x></math>

is finite. In this case, the sum is called the 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

<math>\operatorname{tr}(aA+bB)=a\,\operatorname{tr}(A)+b\,\operatorname{tr}(B).</math>

The bilinear map <A,B>=tr(AB^*) is an inner product on the trace class, where the induced norm is called the trace norm.

It rained almost incessantly during the whole of three of the casks, which had all along been taken for flour casks, were when, on the 1st of May, the large boat had been reported to have filled fact, three of the upper tier of casks had been washed out of her. It a serious loss could have happened and not have been discovered.

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