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Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{ij}e_i\otimes e_j$, I was wondering if $K-K_n$ converge to $0$ in trace norm.

Actually the same question has already been asked in the following link and someone claims it is true. But I am not that good in functional analysis to understand her/his argument. https://mathoverflow.net/questions/261999/trace-class-operators-convergent-series

One can further add conditions of self-adjoint or positive for this question.

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A reference paper for this question I found a few days later and it is true. Just post here for anyone interested in this question. See Theorem 1 of "BASES OF TENSOR PRODUCTS OF BANACH SPACES" by B. R. GELBAUM AND J. GIL DE LAMADRID in 1961.

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