Norm of the upper triangular part of symmetric matrix

No. The best constant is $C_n\sim \ln n$. See, for example, this paper. In particular, for the lower bound, see example 3.3 of the paper.


The $\log n$ result mentioned by Christian Remling is a special case of the results in

Kwapień, S.; Pełczyński, A. The main triangle projection in matrix spaces and its applications. Studia Math. 34 1970 43–68.

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