# Convergence of finite dimensional projection of trace class in trace norm

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.