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.
Verify that $\lVert\cdot\rVert_1$ is a vector norm
0
$\begingroup$
linear-algebra
inequality
vectors
norm
lp-spaces
-
0For some basic information about writing math at this site see e.g. [here](/help/notation), [here](//meta.math.stackexchange.com/q/5020), [here](//meta.stackexchange.com/a/70559) and [here](//meta.math.stackexchange.com/q/1773). – 2017-02-10
1 Answers
1
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?