We are given the 1-norm $$\|x\|_1 = \sum_{i=1}^n |x_i|.$$ We want to show it is a vector norm. It has to satisfy properties
I solved them all except for the last one!
$$\|x+y\|_1 \le \|x\|_1+\|y\|_1$$
Thank you in advance for your help.
2026-04-08 09:16:16.1775639776
Verify that $\lVert\cdot\rVert_1$ is a vector norm <triangular inequality>
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Hint: Use the inequality $|x+y| < |x| + |y|$.
For the proof of the hint visit Sum of absolute values and the absolute value of the sum of these values?