In a book I read about the following definition of convergence in mean squares, in which the $l1$ norm is used as the vector norm. My question is, will the definition have different implications if I use the Euclidean norm instead?
2026-03-25 19:06:33.1774465593
Does the type of norm matter in the definition of convergence in mean square?
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Take a look at norm inequalities. The l1 norm is bigger than the l2 norm, so the theorem in the book implies the convergence with the l2 norm. However, the theorem stated with the l2 norm would not imply that it holds with the l1 norm.