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If $$\|A - x\|_1 \le \epsilon$$ and $$\|B - y\|_1 \le \epsilon$$ where $A, B, x, y \in Herm(H_A)$, where $Herm(H_A)$ are the set of Hermitian matrices in a Hilbert space $H_A$, then can we say, by Triangle Inequality that, $$\|A \otimes B - x \otimes y\|_1 \le 2\epsilon$$ How do I derive it?

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  • $\begingroup$ I've edited the question. Thanks. $\endgroup$ – Hasan Iqbal Mar 13 at 22:50
  • $\begingroup$ What happens when $A$ and $B$ are just real numbers and $x$ and $y$ are both zero? It seems to me that you statement reduces to $|ab| \leq 2 \epsilon$, where I would have expected $\epsilon^2$. Moreover, is there some connection between $A$ and $H_A$? $\endgroup$ – Carl Christian Mar 14 at 10:45

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