*I\'m not sure if I can assume this, but I said that the right half of the inequ

Question

*I'm not sure if I can assume this, but I said that the right half of the inequality is true as the sum of a set of positive elements is always greater than or equal to an individual element within that same set. Then I said that b has to exist as long as it is greater than or equal to 1. Again, not sure if this is sufficient or not. For the left side of the inequality, I am totally lost and I have no idea how to prove that a is the largest and b is the smallest.*

Thanks in advance for your help!

Explanation / Answer

If all values have equal absolute value, the sum can be n times the absolute value of 1 value. Thus, the 1-norm can be n times the infinity-norm, and a = 1/n. (It is easy to show this is the max. Fix the max absolute value and consider any other value for the other coordinates and the value decreases. If there is only 1 non-zero value, then the 1-norm equals the infinity norm, and b = 1. Clearly, it can go lower, as one of the values has to equal the max value. Once again, fix the max-value and consider any other coordinate to have a non-zero value (less than max, of course), and the 1-norm increases while the infinity-norm stays the same, so the 1-norm is more than 1 times the infinity norm. a = 1/n. b = 1.

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