# Triangle Inequality of Tensor Products

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?

• I've edited the question. Thanks. – Hasan Iqbal Mar 13 at 22:50
• 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 $H_A$? – Carl Christian Mar 14 at 10:45